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{{Short description|Continuous function that is not absolutely continuous}}
[[File:CantorEscalier.svg|thumb|right|400px| An approximation to the graph of the Cantor function on the [[unit interval]] ]]
In [[mathematics]], the '''Cantor function''' is an example of a [[function (mathematics)|function]] that is [[continuous function|continuous]], but not [[absolute continuity|absolutely continuous]]. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reaches from 0 to 1. Thus, in one sense the function seems very much like a constant one which cannot grow, and in another, it does indeed monotonically grow, by construction.
It is also referred to as the '''Cantor ternary function''', the '''Lebesgue function''',<ref>{{harvnb|Vestrup|2003|loc=Section 4.6.}}</ref> '''Lebesgue's singular function''', the '''Cantor–Vitali function''', the '''Devil's staircase''',<ref>{{harvnb|Thomson|Bruckner|Bruckner|2008|p=252}}.</ref> the '''Cantor staircase function''',<ref>http://mathworld.wolfram.com/CantorStaircaseFunction.html</ref> and the '''Cantor–Lebesgue function'''.<ref>{{harvnb|Bass|2013|p=28}}.</ref> {{harvs|txt|first=Georg |last=Cantor|authorlink=Georg Cantor|year=1884}} introduced the Cantor function and mentioned that Scheeffer pointed out that it was a [[counterexample]] to an extension of the [[fundamental theorem of calculus]] claimed by [[Carl Gustav Axel Harnack|Harnack]]. The Cantor function was discussed and popularized by {{harvtxt|Scheeffer|1884}}, {{harvtxt|Lebesgue|1904}} and {{harvtxt|Vitali|1905}}.
==Definition==
[[File:Cantor function.gif|300px|right]]
See figure. To formally define the Cantor function ''c'' : [0,1] → [0,1], let ''x'' be in [0,1] and obtain ''c''(''x'') by the following steps:
#Express ''x'' in base 3.
#If ''x'' contains a 1, replace every digit strictly after the first 1 by 0.
#Replace any remaining 2s with 1s.
#Interpret the result as a binary number. The result is ''c''(''x'').
For example:
* 1/4 is 0.02020202... in base 3. There are no 1s so the next stage is still 0.02020202... This is rewritten as 0.01010101... When read in base 2, this corresponds to 1/3, so ''c''(1/4) = 1/3.
* 1/5 is 0.01210121... in base 3. The digits after the first 1 are replaced by 0s to produce 0.01000000... This is not rewritten since there are no 2s. When read in base 2, this corresponds to 1/4, so ''c''(1/5) = 1/4.
* 200/243 is 0.21102 (or 0.211012222...) in base 3. The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. When read in base 2, this corresponds to 3/4, so ''c''(200/243) = 3/4.
Equivalently, if <math>\mathcal{C}</math> is the [[Cantor set]] on [0,1], then the Cantor function ''c''<nowiki> : [0,1] → [0,1] can be defined as</nowiki>
:<math>c(x) =\begin{cases}
\sum_{n=1}^\infty \frac{a_n}{2^n}, & x = \sum_{n=1}^\infty
\frac{2a_n}{3^n}\in\mathcal{C}\ \mathrm{for}\ a_n\in\{0,1\};
\\ \sup_{y\leq x,\, y\in\mathcal{C}} c(y), & x\in [0,1]\setminus \mathcal{C}.\\ \end{cases}
</math>
This formula is well-defined, since every member of the Cantor set has a ''unique'' base 3 representation that only contains the digits 0 or 2. (For some members of <math>\mathcal{C}</math>, the ternary expansion is repeating with trailing 2's and there is an alternative non-repeating expansion ending in 1. For example, 1/3 = 0.1<sub>3</sub> = 0.02222...<sub>3</sub> is a member of the Cantor set). Since ''c''(0) = 0 and ''c''(1) = 1, and ''c'' is monotonic on <math>\mathcal{C}</math>, it is clear that 0 ≤ ''c''(''x'') ≤ 1 also holds for all <math>x\in[0,1]\setminus\mathcal{C}</math>.
==Properties==
The Cantor function challenges naive intuitions about [[continuous function|continuity]] and [[measure (mathematics)|measure]]; though it is continuous everywhere and has zero derivative [[almost everywhere]], <math display="inline">c(x)</math> goes from 0 to 1 as <math display="inline>x</math> goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is [[uniformly continuous]] (precisely, it is [[Hölder continuous]] of exponent ''α'' = log 2/log 3) but not [[absolute continuity|absolutely continuous]]. It is constant on intervals of the form (0.''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>...''x''<sub>n</sub>022222..., 0.''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>...''x''<sub>n</sub>200000...), and every point not in the Cantor set is in one of these intervals, so its derivative is 0 outside of the Cantor set. On the other hand, it has no [[derivative]] at any point in an [[uncountable]] subset of the [[Cantor set]] containing the interval endpoints described above.
The Cantor function can also be seen as the [[cumulative distribution function|cumulative probability distribution function]] of the 1/2-1/2 [[Bernoulli measure]] ''μ'' supported on the Cantor set: <math display="inline">c(x)=\mu([0,x])</math>. This probability distribution, called the [[Cantor distribution]], has no discrete part. That is, the corresponding measure is [[Atom (measure theory)|atomless]]. This is why there are no jump discontinuities in the function; any such jump would correspond to an atom in the measure.
However, no non-constant part of the Cantor function can be represented as an integral of a [[probability density function]]; integrating any putative [[probability density function]] that is not [[almost everywhere]] zero over any interval will give positive probability to some interval to which this distribution assigns probability zero. In particular, as {{harvtxt|Vitali|1905}} pointed out, the function is not the integral of its derivative even though the derivative exists almost everywhere.
The Cantor function is the standard example of a [[singular function]].
The Cantor function is non-decreasing, and so in particular its graph defines a [[rectifiable curve]]. {{harvtxt|Scheeffer|1884}} showed that the arc length of its graph is 2.
===Lack of absolute continuity===
Because the [[Lebesgue measure]] of the [[Uncountable set|uncountably infinite]] [[Cantor set]] is 0, for any positive ''ε'' < 1 and ''δ'', there exists a finite sequence of [[pairwise disjoint]] sub-intervals with total length < ''δ'' over which the Cantor function cumulatively rises more than ''ε''.
In fact, for every ''δ'' > 0 there are finitely many pairwise disjoint intervals (''x<SUB>k</SUB>'',''y<SUB>k</SUB>'') (1 ≤ ''k'' ≤ ''M'') with <math>\sum\limits_{k=1}^M (y_k-x_k)<\delta</math> and <math>\sum\limits_{k=1}^M (c(y_k)-c(x_k))=1</math>.
== Alternative definitions ==
=== Iterative construction ===
[[File:Cantor function sequence.png|250px|right]]
Below we define a sequence {''f''<sub>''n''</sub>} of functions on the unit interval that converges to the Cantor function.
Let ''f''<sub>0</sub>(''x'') = ''x''.
Then, for every integer {{nowrap|''n'' ≥ 0}}, the next function ''f''<sub>''n''+1</sub>(''x'') will be defined in terms of ''f''<sub>''n''</sub>(''x'') as follows:
Let ''f''<sub>''n''+1</sub>(''x'') = {{nowrap|1/2 × ''f''<sub>''n''</sub>(3''x'')}}, when {{nowrap|0 ≤ ''x'' ≤ 1/3 }};
Let ''f''<sub>''n''+1</sub>(''x'') = 1/2, when {{nowrap|1/3 ≤ ''x'' ≤ 2/3 }};
Let ''f''<sub>''n''+1</sub>(''x'') = {{nowrap|1/2 + 1/2 × ''f''<sub>''n''</sub>(3 ''x'' − 2)}}, when {{nowrap|2/3 ≤ ''x'' ≤ 1}}.
The three definitions are compatible at the end-points 1/3 and 2/3, because ''f''<sub>''n''</sub>(0) = 0 and ''f''<sub>''n''</sub>(1) = 1 for every ''n'', by induction. One may check that ''f''<sub>''n''</sub> converges pointwise to the Cantor function defined above. Furthermore, the convergence is uniform. Indeed, separating into three cases, according to the definition of ''f''<sub>''n''+1</sub>, one sees that
:<math>\max_{x \in [0, 1]} |f_{n+1}(x) - f_n(x)| \le \frac 1 2 \, \max_{x \in [0, 1]} |f_{n}(x) - f_{n-1}(x)|, \quad n \ge 1.</math>
If ''f'' denotes the limit function, it follows that, for every ''n'' ≥ 0,
:<math>\max_{x \in [0, 1]} |f(x) - f_n(x)| \le 2^{-n+1} \, \max_{x \in [0, 1]} |f_1(x) - f_0(x)|.</math>
Also the choice of starting function does not really matter, provided ''f''<sub>0</sub>(0) = 0, ''f''<sub>0</sub>(1) = 1 and ''f''<sub>0</sub> is [[Bounded function|bounded]]{{citation needed|date=September 2014}}.
=== Fractal volume ===
The Cantor function is closely related to the [[Cantor set]]. The Cantor set ''C'' can be defined as the set of those numbers in the interval [0, 1] that do not contain the digit 1 in their [[base (exponentiation)|base-3 (triadic) expansion]], except if the 1 is followed by zeros only (in which case the tail 1000<math>\ldots</math> can be replaced by 0222<math>\ldots</math> to get rid of any 1). It turns out that the Cantor set is a [[fractal]] with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume). Only the ''D''-dimensional volume <math> H_D </math> (in the sense of a [[Hausdorff dimension|Hausdorff-measure]]) takes a finite value, where <math> D = \log(2)/\log(3) </math> is the fractal dimension of ''C''. We may define the Cantor function alternatively as the ''D''-dimensional volume of sections of the Cantor set
: <math>
f(x)=H_D(C \cap (0,x)).
</math>
==Self-similarity==
The Cantor function possesses several [[symmetry|symmetries]]. For <math>0\le x\le 1</math>, there is a reflection symmetry
:<math>c(x)=1-c(1-x)</math>
and a pair of magnifications, one on the left and one on the right:
:<math>c\left(\frac{x}{3}\right) = \frac{c(x)}{2}</math>
and
:<math>c\left(\frac{x+2}{3}\right) = \frac{1+c(x)}{2}</math>
The magnifications can be cascaded; they generate the [[dyadic monoid]]. This is exhibited by defining several helper functions. Define the reflection as
:<math>r(x)=1-x</math>
The first self-symmetry can be expressed as
:<math>r\circ c = c\circ r</math>
where the symbol <math>\circ</math> denotes function composition. That is, <math>(r\circ c)(x)=r(c(x))=1-c(x)</math> and likewise for the other cases. For the left and right magnifications, write the left-mappings
:<math>L_D(x)= \frac{x}{2}</math> and <math>L_C(x)= \frac{x}{3}</math>
Then the Cantor function obeys
:<math>L_D \circ c = c \circ L_C</math>
Similarly, define the right mappings as
:<math>R_D(x)= \frac{1+x}{2}</math> and <math>R_C(x)= \frac{2+x}{3}</math>
Then, likewise,
:<math>R_D \circ c = c \circ R_C</math>
The two sides can be mirrored one onto the other, in that
:<math>L_D \circ r = r\circ R_D</math>
and likewise,
:<math>L_C \circ r = r\circ R_C</math>
These operations can be stacked arbitrarily. Consider, for example, the sequence of left-right moves <math>LRLLR.</math> Adding the subscripts C and D, and, for clarity, dropping the composition operator <math>\circ</math> in all but a few places, one has:
:<math>L_D R_D L_D L_D R_D \circ c = c \circ L_C R_C L_C L_C R_C</math>
Arbitrary finite-length strings in the letters L and R correspond to the [[dyadic rationals]], in that every dyadic rational can be written as both <math>y=n/2^m</math> for integer ''n'' and ''m'' and as finite length of bits <math>y=0.b_1b_2b_3\cdots b_m</math> with <math>b_k\in \{0,1\}.</math> Thus, every dyadic rational is in one-to-one correspondence with some self-symmetry of the Cantor function.
Some notational rearrangements can make the above slightly easier to express. Let <math>g_0</math> and <math>g_1</math> stand for L and R. Function composition extends this to a [[monoid]], in that one can write <math>g_{010}=g_0g_1g_0</math> and generally, <math>g_Ag_B=g_{AB}</math> for some binary strings of digits ''A'', ''B'', where ''AB'' is just the ordinary [[concatenation]] of such strings. The dyadic monoid ''M'' is then the monoid of all such finite-length left-right moves. Writing <math>\gamma\in M</math> as a general element of the monoid, there is a corresponding self-symmetry of the Cantor function:
:<math>\gamma_D\circ c= c\circ \gamma_C</math>
The dyadic monoid itself has several interesting properties. It can be viewed as a finite number of left-right moves down an infinite [[binary tree]]; the infinitely distant "leaves" on the tree correspond to the points on the Cantor set, and so, the monoid also represents the self-symmetries of the Cantor set. In fact, a large class of commonly occurring fractals are described by the dyadic monoid; additional examples can be found in the article on [[de Rham curve]]s. Other fractals possessing self-similarity are described with other kinds of monoids. The dyadic monoid is itself a sub-monoid of the [[modular group]] <math>SL(2,\mathbb{Z}).</math>
Note that the Cantor function bears more than a passing resemblance to [[Minkowski's question-mark function]]. In particular, it obeys the exact same symmetry relations, although in an altered form.
== Generalizations ==
Let
: <math>y=\sum_{k=1}^\infty b_k 2^{-k}</math>
be the [[dyadic rational|dyadic]] (binary) expansion of the real number 0 ≤ ''y'' ≤ 1 in terms of binary digits ''b''<sub>''k''</sub> ∈ {0,1}. This expansion is discussed in greater detail in the article on the [[dyadic transformation]]. Then consider the function
: <math>C_z(y)=\sum_{k=1}^\infty b_k z^{k}.</math>
For ''z'' = 1/3, the inverse of the function ''x'' = 2 ''C''<sub>1/3</sub>(''y'') is the Cantor function. That is, ''y'' = ''y''(''x'') is the Cantor function. In general, for any ''z'' < 1/2, ''C''<sub>''z''</sub>(''y'') looks like the Cantor function turned on its side, with the width of the steps getting wider as ''z'' approaches zero.
As mentioned above, the Cantor function is also the cumulative distribution function of a measure on the Cantor set. Different Cantor functions, or Devil's Staircases, can be obtained by considering different atom-less probability measures supported on the Cantor set or other fractals. While the Cantor function has derivative 0 almost everywhere, current research focusses on the question of the size of the set of points where the upper right derivative is distinct from the lower right derivative, causing the derivative to not exist. This analysis of differentiability is usually given in terms of [[fractal dimension]], with the Hausdorff dimension the most popular choice. This line of research was started in the 1990s by Darst,<ref>{{Cite journal|title = The Hausdorff Dimension of the Nondifferentiability Set of the Cantor Function is [ ln(2)/ln(3) ]2|jstor = 2159830|journal = Proceedings of the American Mathematical Society|date = 1993-09-01|pages = 105–108|volume = 119|issue = 1|doi = 10.2307/2159830|first = Richard|last = Darst}}</ref> who showed that the Hausdorff dimension of the set of non-differentiability of the Cantor function is the square of the dimension of the Cantor set, <math>(\log2/\log3)^2</math>. Subsequently [[Kenneth Falconer (mathematician)|Falconer]]<ref>{{Cite journal|title = One-sided multifractal analysis and points of non-differentiability of devil's staircases|journal = Mathematical Proceedings of the Cambridge Philosophical Society|date = 2004-01-01|issn = 1469-8064|pages = 167–174|volume = 136|issue = 1|doi = 10.1017/S0305004103006960|first = Kenneth J.|last = Falconer|bibcode = 2004MPCPS.136..167F}}</ref> showed that this squaring relationship holds for all Ahlfor's regular, singular measures, i.e.<math display="block">\dim_H\left\{x : f'(x)=\lim_{h\to0^+}\frac{\mu([x,x+h])}{h}\text{ does not exist}\right\}=\left(\dim_H\operatorname{supp}(\mu)\right)^2</math>Later, Troscheit<ref>{{Cite journal|title = Hölder differentiability of self-conformal devil's staircases|journal = Mathematical Proceedings of the Cambridge Philosophical Society|date = 2014-03-01|issn = 1469-8064|pages = 295–311|volume = 156|issue = 2|doi = 10.1017/S0305004113000698|first = Sascha|last = Troscheit|arxiv = 1301.1286|bibcode = 2014MPCPS.156..295T|s2cid = 56402751}}</ref> obtain a more comprehensive picture of the set where the derivative does not exist for more general normalized Gibb's measures supported on self-conformal and [[Self-similarity|self-similar sets]].
[[Hermann Minkowski]]'s [[Minkowski's question mark function|question mark function]] loosely resembles the Cantor function visually, appearing as a "smoothed out" form of the latter; it can be constructed by passing from a continued fraction expansion to a binary expansion, just as the Cantor function can be constructed by passing from a ternary expansion to a binary expansion. The question mark function has the interesting property of having vanishing derivatives at all rational numbers.
== See also==
* [[Dyadic transformation]]
==Notes==
<references />
==References==
* {{cite book|first1=Richard Franklin|last1=Bass|author1-link=Richard F. Bass|title=Real analysis for graduate students|year=2013|orig-year=2011|edition=Second|publisher=Createspace Independent Publishing|isbn=978-1-4818-6914-0}}
*{{cite journal | last=Cantor | first=G. | title=De la puissance des ensembles parfaits de points: Extrait d'une lettre adressée à l'éditeur | journal=Acta Mathematica | publisher=International Press of Boston | volume=4 | year=1884 | issn=0001-5962 | doi=10.1007/bf02418423 | pages=381–392 |trans-title=The power of perfect sets of points: Extract from a letter addressed to the editor| doi-access=free }} Reprinted in: E. Zermelo (Ed.), Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts, Springer, New York, 1980.
*{{citation|mr=2681574
|last1=Darst|first1= Richard B.|last2= Palagallo|first2= Judith A.|last3= Price|first3= Thomas E.
|title=Curious curves|publisher= World Scientific Publishing Co. Pte. Ltd.|place= Hackensack, NJ |year=2010|isbn= 978-981-4291-28-6}}
*{{cite journal | last1=Dovgoshey | first1=O. | last2=Martio | first2=O. | last3=Ryazanov | first3=V. | last4=Vuorinen | first4=M. | title=The Cantor function | journal=Expositiones Mathematicae | publisher=Elsevier BV | volume=24 | issue=1 | year=2006 | issn=0723-0869 | doi=10.1016/j.exmath.2005.05.002 | pages=1–37 |mr=2195181 |url=http://users.utu.fi/vuorinen/REA12/107.pdf}}
*{{cite journal | last=Fleron | first=Julian F. | title=A Note on the History of the Cantor Set and Cantor Function | journal=Mathematics Magazine | publisher=Informa UK Limited | volume=67 | issue=2 | pages=136–140 | date=1994-04-01 | issn=0025-570X | doi=10.2307/2690689 |jstor=2690689}}
*{{citation|first=H.|last= Lebesgue |title=Leçons sur l'intégration et la recherche des fonctions primitives|place= Paris|publisher= Gauthier-Villars|year= 1904 |trans-title=Lessons on integration and search for primitive functions}}
*{{cite book | last=Leoni | first=Giovanni | title=A first course in Sobolev spaces | publisher=American Mathematical Society | ___location=Providence, Rhode Island | year=2017 | isbn=978-1-4704-2921-8 | oclc=976406106 | page=734 |edition=2nd |url=http://bookstore.ams.org/gsm-181/ |volume=181}}
*{{cite journal | last=Scheeffer | first=Ludwig | title=Allgemeine Untersuchungen über Rectification der Curven | journal=Acta Mathematica | publisher=International Press of Boston | volume=5 | year=1884 | issn=0001-5962 | doi=10.1007/bf02421552 | pages=49–82 |trans-title=General investigations on rectification of the curves| doi-access=free }}
* {{cite book|last1=Thomson|first1=Brian S.|last2=Bruckner|first2=Judith B.|last3=Bruckner|first3=Andrew M.|title=Elementary real analysis|publisher=ClassicalRealAnalysis.com|edition=Second|year=2008|orig-year=2001|isbn=978-1-4348-4367-8}}
*{{cite book|title=The theory of measures and integration|last=Vestrup|first=E.M.|series=Wiley series in probability and statistics|publisher=John Wiley & sons|year=2003|isbn=978-0471249771}}
*{{citation|first=A.|last= Vitali|title=Sulle funzioni integrali|journal=Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.|volume= 40 |year=1905 |pages= 1021–1034 |trans-title=On the integral functions}}
== External links ==
* [https://www.encyclopediaofmath.org/index.php/Cantor_ternary_function ''Cantor ternary function'' at Encyclopaedia of Mathematics]
* [http://demonstrations.wolfram.com/CantorFunction/ Cantor Function] by Douglas Rivers, the [[Wolfram Demonstrations Project]].
* {{MathWorld |title= Cantor Function |urlname= CantorFunction}}
[[Category:Fractals]]
[[Category:Measure theory]]
[[Category:Special functions]]
[[Category:Georg Cantor]]
[[Category:De Rham curves]]' |
New page wikitext, after the edit (new_wikitext) | '{{Use American English|date = March 2019}}
{{Short description|Continuous function that is not absolutely continuous}}
[[File:CantorEscalier.svg|thumb|right|400px| An approximation to the graph of the Cantor function on the [[unit interval]] ]]
In [[mathematics]], the '''Cantor function''' is an example of a [[function (mathematics)|function]] that is [[continuous function|continuous]], but not [[absolute continuity|absolutely continuous]]. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reachmath> can be replaced by 0222<math>\ldots</math> to get rid of any 1). It turns out that the Cantor set is a [[fractal]] with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume). Only the ''D''-dimensional volume <math> H_D </math> (in the sense of a [[Hausdorff dimension|Hausdorff-measure]]) takes a finite value, where <math> D = \log(2)/\log(3) </math> is the fractal dimension of ''C''. We may define the Cantor function alternatively as the ''D''-dimensional volume of sections of the Cantor set
: <math>
f(x)=H_D(C \cap (0,x)).
</math>
==Self-similarity==
The Cantor function possesses several [[symmetry|symmetries]]. For <math>0\le x\le 1</math>, there is a reflection symmetry
:<math>c(x)=1-c(1-x)</math>
and a pair of magnifications, one on the left and one on the right:
:<math>c\left(\frac{x}{3}\right) = \frac{c(x)}{2}</math>
and
:<math>c\left(\frac{x+2}{3}\right) = \frac{1+c(x)}{2}</math>
The magnifications can be cascaded; they generate the [[dyadic monoid]]. This is exhibited by defining several helper functions. Define the reflection as
:<math>r(x)=1-x</math>
The first self-symmetry can be expressed as
:<math>r\circ c = c\circ r</math>
where the symbol <math>\circ</math> denotes function composition. That is, <math>(r\circ c)(x)=r(c(x))=1-c(x)</math> and likewise for the other cases. For the left and right magnifications, write the left-mappings
:<math>L_D(x)= \frac{x}{2}</math> and <math>L_C(x)= \frac{x}{3}</math>
Then the Cantor function obeys
:<math>L_D \circ c = c \circ L_C</math>
Similarly, define the right mappings as
:<math>R_D(x)= \frac{1+x}{2}</math> and <math>R_C(x)= \frac{2+x}{3}</math>
Then, likewise,
:<math>R_D \circ c = c \circ R_C</math>
The two sides can be mirrored one onto the other, in that
:<math>L_D \circ r = r\circ R_D</math>
and likewise,
:<math>L_C \circ r = r\circ R_C</math>
These operations can be stacked arbitrarily. Consider, for example, the sequence of left-right moves <math>LRLLR.</math> Adding the subscripts C and D, and, for clarity, dropping the composition operator <math>\circ</math> in all but a few places, one has:
:<math>L_D R_D L_D L_D R_D \circ c = c \circ L_C R_C L_C L_C R_C</math>
Arbitrary finite-length strings in the letters L and R correspond to the [[dyadic rationals]], in that every dyadic rational can be written as both <math>y=n/2^m</math> for integer ''n'' and ''m'' and as finite length of bits <math>y=0.b_1b_2b_3\cdots b_m</math> with <math>b_k\in \{0,1\}.</math> Thus, every dyadic rational is in one-to-one correspondence with some self-symmetry of the Cantor function.
Some notational rearrangements can make the above slightly easier to express. Let <math>g_0</math> and <math>g_1</math> stand for L and R. Function composition extends this to a [[monoid]], in that one can write <math>g_{010}=g_0g_1g_0</math> and generally, <math>g_Ag_B=g_{AB}</math> for some binary strings of digits ''A'', ''B'', where ''AB'' is just the ordinary [[concatenation]] of such strings. The dyadic monoid ''M'' is then the monoid of all such finite-length left-right moves. Writing <math>\gamma\in M</math> as a general element of the monoid, there is a corresponding self-symmetry of the Cantor function:
:<math>\gamma_D\circ c= c\circ \gamma_C</math>
The dyadic monoid itself has several interesting properties. It can be viewed as a finite number of left-right moves down an infinite [[binary tree]]; the infinitely distant "leaves" on the tree correspond to the points on the Cantor set, and so, the monoid also represents the self-symmetries of the Cantor set. In fact, a large class of commonly occurring fractals are described by the dyadic monoid; additional examples can be found in the article on [[de Rham curve]]s. Other fractals possessing self-similarity are described with other kinds of monoids. The dyadic monoid is itself a sub-monoid of the [[modular group]] <math>SL(2,\mathbb{Z}).</math>
Note that the Cantor function bears more than a passing resemblance to [[Minkowski's question-mark function]]. In particular, it obeys the exact same symmetry relations, although in an altered form.
== Generalizations ==
Let
: <math>y=\sum_{k=1}^\infty b_k 2^{-k}</math>
be the [[dyadic rational|dyadic]] (binary) expansion of the real number 0 ≤ ''y'' ≤ 1 in terms of binary digits ''b''<sub>''k''</sub> ∈ {0,1}. This expansion is discussed in greater detail in the article on the [[dyadic transformation]]. Then consider the function
: <math>C_z(y)=\sum_{k=1}^\infty b_k z^{k}.</math>
For ''z'' = 1/3, the inverse of the function ''x'' = 2 ''C''<sub>1/3</sub>(''y'') is the Cantor function. That is, ''y'' = ''y''(''x'') is the Cantor function. In general, for any ''z'' < 1/2, ''C''<sub>''z''</sub>(''y'') looks like the Cantor function turned on its side, with the width of the steps getting wider as ''z'' approaches zero.
As mentioned above, the Cantor function is also the cumulative distribution function of a measure on the Cantor set. Different Cantor functions, or Devil's Staircases, can be obtained by considering different atom-less probability measures supported on the Cantor set or other fractals. While the Cantor function has derivative 0 almost everywhere, current research focusses on the question of the size of the set of points where the upper right derivative is distinct from the lower right derivative, causing the derivative to not exist. This analysis of differentiability is usually given in terms of [[fractal dimension]], with the Hausdorff dimension the most popular choice. This line of research was started in the 1990s by Darst,<ref>{{Cite journal|title = The Hausdorff Dimension of the Nondifferentiability Set of the Cantor Function is [ ln(2)/ln(3) ]2|jstor = 2159830|journal = Proceedings of the American Mathematical Society|date = 1993-09-01|pages = 105–108|volume = 119|issue = 1|doi = 10.2307/2159830|first = Richard|last = Darst}}</ref> who showed that the Hausdorff dimension of the set of non-differentiability of the Cantor function is the square of the dimension of the Cantor set, <math>(\log2/\log3)^2</math>. Subsequently [[Kenneth Falconer (mathematician)|Falconer]]<ref>{{Cite journal|title = One-sided multifractal analysis and points of non-differentiability of devil's staircases|journal = Mathematical Proceedings of the Cambridge Philosophical Society|date = 2004-01-01|issn = 1469-8064|pages = 167–174|volume = 136|issue = 1|doi = 10.1017/S0305004103006960|first = Kenneth J.|last = Falconer|bibcode = 2004MPCPS.136..167F}}</ref> showed that this squaring relationship holds for all Ahlfor's regular, singular measures, i.e.<math display="block">\dim_H\left\{x : f'(x)=\lim_{h\to0^+}\frac{\mu([x,x+h])}{h}\text{ does not exist}\right\}=\left(\dim_H\operatorname{supp}(\mu)\right)^2</math>Later, Troscheit<ref>{{Cite journal|title = Hölder differentiability of self-conformal devil's staircases|journal = Mathematical Proceedings of the Cambridge Philosophical Society|date = 2014-03-01|issn = 1469-8064|pages = 295–311|volume = 156|issue = 2|doi = 10.1017/S0305004113000698|first = Sascha|last = Troscheit|arxiv = 1301.1286|bibcode = 2014MPCPS.156..295T|s2cid = 56402751}}</ref> obtain a more comprehensive picture of the set where the derivative does not exist for more general normalized Gibb's measures supported on self-conformal and [[Self-similarity|self-similar sets]].
[[Hermann Minkowski]]'s [[Minkowski's question mark function|question mark function]] loosely resembles the Cantor function visually, appearing as a "smoothed out" form of the latter; it can be constructed by passing from a continued fraction expansion to a binary expansion, just as the Cantor function can be constructed by passing from a ternary expansion to a binary expansion. The question mark function has the interesting property of having vanishing derivatives at all rational numbers.
== See also==
* [[Dyadic transformation]]
==Notes==
<references />
==References==
* {{cite book|first1=Richard Franklin|last1=Bass|author1-link=Richard F. Bass|title=Real analysis for graduate students|year=2013|orig-year=2011|edition=Second|publisher=Createspace Independent Publishing|isbn=978-1-4818-6914-0}}
*{{cite journal | last=Cantor | first=G. | title=De la puissance des ensembles parfaits de points: Extrait d'une lettre adressée à l'éditeur | journal=Acta Mathematica | publisher=International Press of Boston | volume=4 | year=1884 | issn=0001-5962 | doi=10.1007/bf02418423 | pages=381–392 |trans-title=The power of perfect sets of points: Extract from a letter addressed to the editor| doi-access=free }} Reprinted in: E. Zermelo (Ed.), Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts, Springer, New York, 1980.
*{{citation|mr=2681574
|last1=Darst|first1= Richard B.|last2= Palagallo|first2= Judith A.|last3= Price|first3= Thomas E.
|title=Curious curves|publisher= World Scientific Publishing Co. Pte. Ltd.|place= Hackensack, NJ |year=2010|isbn= 978-981-4291-28-6}}
*{{cite journal | last1=Dovgoshey | first1=O. | last2=Martio | first2=O. | last3=Ryazanov | first3=V. | last4=Vuorinen | first4=M. | title=The Cantor function | journal=Expositiones Mathematicae | publisher=Elsevier BV | volume=24 | issue=1 | year=2006 | issn=0723-0869 | doi=10.1016/j.exmath.2005.05.002 | pages=1–37 |mr=2195181 |url=http://users.utu.fi/vuorinen/REA12/107.pdf}}
*{{cite journal | last=Fleron | first=Julian F. | title=A Note on the History of the Cantor Set and Cantor Function | journal=Mathematics Magazine | publisher=Informa UK Limited | volume=67 | issue=2 | pages=136–140 | date=1994-04-01 | issn=0025-570X | doi=10.2307/2690689 |jstor=2690689}}
*{{citation|first=H.|last= Lebesgue |title=Leçons sur l'intégration et la recherche des fonctions primitives|place= Paris|publisher= Gauthier-Villars|year= 1904 |trans-title=Lessons on integration and search for primitive functions}}
*{{cite book | last=Leoni | first=Giovanni | title=A first course in Sobolev spaces | publisher=American Mathematical Society | ___location=Providence, Rhode Island | year=2017 | isbn=978-1-4704-2921-8 | oclc=976406106 | page=734 |edition=2nd |url=http://bookstore.ams.org/gsm-181/ |volume=181}}
*{{cite journal | last=Scheeffer | first=Ludwig | title=Allgemeine Untersuchungen über Rectification der Curven | journal=Acta Mathematica | publisher=International Press of Boston | volume=5 | year=1884 | issn=0001-5962 | doi=10.1007/bf02421552 | pages=49–82 |trans-title=General investigations on rectification of the curves| doi-access=free }}
* {{cite book|last1=Thomson|first1=Brian S.|last2=Bruckner|first2=Judith B.|last3=Bruckner|first3=Andrew M.|title=Elementary real analysis|publisher=ClassicalRealAnalysis.com|edition=Second|year=2008|orig-year=2001|isbn=978-1-4348-4367-8}}
*{{cite book|title=The theory of measures and integration|last=Vestrup|first=E.M.|series=Wiley series in probability and statistics|publisher=John Wiley & sons|year=2003|isbn=978-0471249771}}
*{{citation|first=A.|last= Vitali|title=Sulle funzioni integrali|journal=Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.|volume= 40 |year=1905 |pages= 1021–1034 |trans-title=On the integral functions}}
== External links ==
* [https://www.encyclopediaofmath.org/index.php/Cantor_ternary_function ''Cantor ternary function'' at Encyclopaedia of Mathematics]
* [http://demonstrations.wolfram.com/CantorFunction/ Cantor Function] by Douglas Rivers, the [[Wolfram Demonstrations Project]].
* {{MathWorld |title= Cantor Function |urlname= CantorFunction}}
[[Category:Fractals]]
[[Category:Measure theory]]
[[Category:Special functions]]
[[Category:Georg Cantor]]
[[Category:De Rham curves]]' |
Unified diff of changes made by edit (edit_diff) | '@@ -2,78 +2,5 @@
{{Short description|Continuous function that is not absolutely continuous}}
[[File:CantorEscalier.svg|thumb|right|400px| An approximation to the graph of the Cantor function on the [[unit interval]] ]]
-In [[mathematics]], the '''Cantor function''' is an example of a [[function (mathematics)|function]] that is [[continuous function|continuous]], but not [[absolute continuity|absolutely continuous]]. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reaches from 0 to 1. Thus, in one sense the function seems very much like a constant one which cannot grow, and in another, it does indeed monotonically grow, by construction.
-
-It is also referred to as the '''Cantor ternary function''', the '''Lebesgue function''',<ref>{{harvnb|Vestrup|2003|loc=Section 4.6.}}</ref> '''Lebesgue's singular function''', the '''Cantor–Vitali function''', the '''Devil's staircase''',<ref>{{harvnb|Thomson|Bruckner|Bruckner|2008|p=252}}.</ref> the '''Cantor staircase function''',<ref>http://mathworld.wolfram.com/CantorStaircaseFunction.html</ref> and the '''Cantor–Lebesgue function'''.<ref>{{harvnb|Bass|2013|p=28}}.</ref> {{harvs|txt|first=Georg |last=Cantor|authorlink=Georg Cantor|year=1884}} introduced the Cantor function and mentioned that Scheeffer pointed out that it was a [[counterexample]] to an extension of the [[fundamental theorem of calculus]] claimed by [[Carl Gustav Axel Harnack|Harnack]]. The Cantor function was discussed and popularized by {{harvtxt|Scheeffer|1884}}, {{harvtxt|Lebesgue|1904}} and {{harvtxt|Vitali|1905}}.
-
-==Definition==
-[[File:Cantor function.gif|300px|right]]
-
-See figure. To formally define the Cantor function ''c'' : [0,1] → [0,1], let ''x'' be in [0,1] and obtain ''c''(''x'') by the following steps:
-
-#Express ''x'' in base 3.
-#If ''x'' contains a 1, replace every digit strictly after the first 1 by 0.
-#Replace any remaining 2s with 1s.
-#Interpret the result as a binary number. The result is ''c''(''x'').
-
-For example:
-* 1/4 is 0.02020202... in base 3. There are no 1s so the next stage is still 0.02020202... This is rewritten as 0.01010101... When read in base 2, this corresponds to 1/3, so ''c''(1/4) = 1/3.
-* 1/5 is 0.01210121... in base 3. The digits after the first 1 are replaced by 0s to produce 0.01000000... This is not rewritten since there are no 2s. When read in base 2, this corresponds to 1/4, so ''c''(1/5) = 1/4.
-* 200/243 is 0.21102 (or 0.211012222...) in base 3. The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. When read in base 2, this corresponds to 3/4, so ''c''(200/243) = 3/4.
-
-Equivalently, if <math>\mathcal{C}</math> is the [[Cantor set]] on [0,1], then the Cantor function ''c''<nowiki> : [0,1] → [0,1] can be defined as</nowiki>
-
-:<math>c(x) =\begin{cases}
- \sum_{n=1}^\infty \frac{a_n}{2^n}, & x = \sum_{n=1}^\infty
-\frac{2a_n}{3^n}\in\mathcal{C}\ \mathrm{for}\ a_n\in\{0,1\};
-\\ \sup_{y\leq x,\, y\in\mathcal{C}} c(y), & x\in [0,1]\setminus \mathcal{C}.\\ \end{cases}
-</math>
-
-This formula is well-defined, since every member of the Cantor set has a ''unique'' base 3 representation that only contains the digits 0 or 2. (For some members of <math>\mathcal{C}</math>, the ternary expansion is repeating with trailing 2's and there is an alternative non-repeating expansion ending in 1. For example, 1/3 = 0.1<sub>3</sub> = 0.02222...<sub>3</sub> is a member of the Cantor set). Since ''c''(0) = 0 and ''c''(1) = 1, and ''c'' is monotonic on <math>\mathcal{C}</math>, it is clear that 0 ≤ ''c''(''x'') ≤ 1 also holds for all <math>x\in[0,1]\setminus\mathcal{C}</math>.
-
-==Properties==
-The Cantor function challenges naive intuitions about [[continuous function|continuity]] and [[measure (mathematics)|measure]]; though it is continuous everywhere and has zero derivative [[almost everywhere]], <math display="inline">c(x)</math> goes from 0 to 1 as <math display="inline>x</math> goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is [[uniformly continuous]] (precisely, it is [[Hölder continuous]] of exponent ''α'' = log 2/log 3) but not [[absolute continuity|absolutely continuous]]. It is constant on intervals of the form (0.''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>...''x''<sub>n</sub>022222..., 0.''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>...''x''<sub>n</sub>200000...), and every point not in the Cantor set is in one of these intervals, so its derivative is 0 outside of the Cantor set. On the other hand, it has no [[derivative]] at any point in an [[uncountable]] subset of the [[Cantor set]] containing the interval endpoints described above.
-
-The Cantor function can also be seen as the [[cumulative distribution function|cumulative probability distribution function]] of the 1/2-1/2 [[Bernoulli measure]] ''μ'' supported on the Cantor set: <math display="inline">c(x)=\mu([0,x])</math>. This probability distribution, called the [[Cantor distribution]], has no discrete part. That is, the corresponding measure is [[Atom (measure theory)|atomless]]. This is why there are no jump discontinuities in the function; any such jump would correspond to an atom in the measure.
-
-However, no non-constant part of the Cantor function can be represented as an integral of a [[probability density function]]; integrating any putative [[probability density function]] that is not [[almost everywhere]] zero over any interval will give positive probability to some interval to which this distribution assigns probability zero. In particular, as {{harvtxt|Vitali|1905}} pointed out, the function is not the integral of its derivative even though the derivative exists almost everywhere.
-
-The Cantor function is the standard example of a [[singular function]].
-
-The Cantor function is non-decreasing, and so in particular its graph defines a [[rectifiable curve]]. {{harvtxt|Scheeffer|1884}} showed that the arc length of its graph is 2.
-
-===Lack of absolute continuity===
-Because the [[Lebesgue measure]] of the [[Uncountable set|uncountably infinite]] [[Cantor set]] is 0, for any positive ''ε'' < 1 and ''δ'', there exists a finite sequence of [[pairwise disjoint]] sub-intervals with total length < ''δ'' over which the Cantor function cumulatively rises more than ''ε''.
-
-In fact, for every ''δ'' > 0 there are finitely many pairwise disjoint intervals (''x<SUB>k</SUB>'',''y<SUB>k</SUB>'') (1 ≤ ''k'' ≤ ''M'') with <math>\sum\limits_{k=1}^M (y_k-x_k)<\delta</math> and <math>\sum\limits_{k=1}^M (c(y_k)-c(x_k))=1</math>.
-
-== Alternative definitions ==
-
-=== Iterative construction ===
-[[File:Cantor function sequence.png|250px|right]]
-
-Below we define a sequence {''f''<sub>''n''</sub>} of functions on the unit interval that converges to the Cantor function.
-
-Let ''f''<sub>0</sub>(''x'') = ''x''.
-
-Then, for every integer {{nowrap|''n'' ≥ 0}}, the next function ''f''<sub>''n''+1</sub>(''x'') will be defined in terms of ''f''<sub>''n''</sub>(''x'') as follows:
-
-Let ''f''<sub>''n''+1</sub>(''x'') = {{nowrap|1/2 × ''f''<sub>''n''</sub>(3''x'')}}, when {{nowrap|0 ≤ ''x'' ≤ 1/3 }};
-
-Let ''f''<sub>''n''+1</sub>(''x'') = 1/2, when {{nowrap|1/3 ≤ ''x'' ≤ 2/3 }};
-
-Let ''f''<sub>''n''+1</sub>(''x'') = {{nowrap|1/2 + 1/2 × ''f''<sub>''n''</sub>(3 ''x'' − 2)}}, when {{nowrap|2/3 ≤ ''x'' ≤ 1}}.
-
-The three definitions are compatible at the end-points 1/3 and 2/3, because ''f''<sub>''n''</sub>(0) = 0 and ''f''<sub>''n''</sub>(1) = 1 for every ''n'', by induction. One may check that ''f''<sub>''n''</sub> converges pointwise to the Cantor function defined above. Furthermore, the convergence is uniform. Indeed, separating into three cases, according to the definition of ''f''<sub>''n''+1</sub>, one sees that
-
-:<math>\max_{x \in [0, 1]} |f_{n+1}(x) - f_n(x)| \le \frac 1 2 \, \max_{x \in [0, 1]} |f_{n}(x) - f_{n-1}(x)|, \quad n \ge 1.</math>
-
-If ''f'' denotes the limit function, it follows that, for every ''n'' ≥ 0,
-
-:<math>\max_{x \in [0, 1]} |f(x) - f_n(x)| \le 2^{-n+1} \, \max_{x \in [0, 1]} |f_1(x) - f_0(x)|.</math>
-
-Also the choice of starting function does not really matter, provided ''f''<sub>0</sub>(0) = 0, ''f''<sub>0</sub>(1) = 1 and ''f''<sub>0</sub> is [[Bounded function|bounded]]{{citation needed|date=September 2014}}.
-
-=== Fractal volume ===
-The Cantor function is closely related to the [[Cantor set]]. The Cantor set ''C'' can be defined as the set of those numbers in the interval [0, 1] that do not contain the digit 1 in their [[base (exponentiation)|base-3 (triadic) expansion]], except if the 1 is followed by zeros only (in which case the tail 1000<math>\ldots</math> can be replaced by 0222<math>\ldots</math> to get rid of any 1). It turns out that the Cantor set is a [[fractal]] with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume). Only the ''D''-dimensional volume <math> H_D </math> (in the sense of a [[Hausdorff dimension|Hausdorff-measure]]) takes a finite value, where <math> D = \log(2)/\log(3) </math> is the fractal dimension of ''C''. We may define the Cantor function alternatively as the ''D''-dimensional volume of sections of the Cantor set
+In [[mathematics]], the '''Cantor function''' is an example of a [[function (mathematics)|function]] that is [[continuous function|continuous]], but not [[absolute continuity|absolutely continuous]]. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reachmath> can be replaced by 0222<math>\ldots</math> to get rid of any 1). It turns out that the Cantor set is a [[fractal]] with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume). Only the ''D''-dimensional volume <math> H_D </math> (in the sense of a [[Hausdorff dimension|Hausdorff-measure]]) takes a finite value, where <math> D = \log(2)/\log(3) </math> is the fractal dimension of ''C''. We may define the Cantor function alternatively as the ''D''-dimensional volume of sections of the Cantor set
: <math>
' |
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0 => 'In [[mathematics]], the '''Cantor function''' is an example of a [[function (mathematics)|function]] that is [[continuous function|continuous]], but not [[absolute continuity|absolutely continuous]]. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reachmath> can be replaced by 0222<math>\ldots</math> to get rid of any 1). It turns out that the Cantor set is a [[fractal]] with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume). Only the ''D''-dimensional volume <math> H_D </math> (in the sense of a [[Hausdorff dimension|Hausdorff-measure]]) takes a finite value, where <math> D = \log(2)/\log(3) </math> is the fractal dimension of ''C''. We may define the Cantor function alternatively as the ''D''-dimensional volume of sections of the Cantor set'
] |
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0 => 'In [[mathematics]], the '''Cantor function''' is an example of a [[function (mathematics)|function]] that is [[continuous function|continuous]], but not [[absolute continuity|absolutely continuous]]. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reaches from 0 to 1. Thus, in one sense the function seems very much like a constant one which cannot grow, and in another, it does indeed monotonically grow, by construction.',
1 => '',
2 => 'It is also referred to as the '''Cantor ternary function''', the '''Lebesgue function''',<ref>{{harvnb|Vestrup|2003|loc=Section 4.6.}}</ref> '''Lebesgue's singular function''', the '''Cantor–Vitali function''', the '''Devil's staircase''',<ref>{{harvnb|Thomson|Bruckner|Bruckner|2008|p=252}}.</ref> the '''Cantor staircase function''',<ref>http://mathworld.wolfram.com/CantorStaircaseFunction.html</ref> and the '''Cantor–Lebesgue function'''.<ref>{{harvnb|Bass|2013|p=28}}.</ref> {{harvs|txt|first=Georg |last=Cantor|authorlink=Georg Cantor|year=1884}} introduced the Cantor function and mentioned that Scheeffer pointed out that it was a [[counterexample]] to an extension of the [[fundamental theorem of calculus]] claimed by [[Carl Gustav Axel Harnack|Harnack]]. The Cantor function was discussed and popularized by {{harvtxt|Scheeffer|1884}}, {{harvtxt|Lebesgue|1904}} and {{harvtxt|Vitali|1905}}.',
3 => '',
4 => '==Definition==',
5 => '[[File:Cantor function.gif|300px|right]]',
6 => '',
7 => 'See figure. To formally define the Cantor function ''c'' : [0,1] → [0,1], let ''x'' be in [0,1] and obtain ''c''(''x'') by the following steps:',
8 => '',
9 => '#Express ''x'' in base 3. ',
10 => '#If ''x'' contains a 1, replace every digit strictly after the first 1 by 0.',
11 => '#Replace any remaining 2s with 1s.',
12 => '#Interpret the result as a binary number. The result is ''c''(''x'').',
13 => '',
14 => 'For example:',
15 => '* 1/4 is 0.02020202... in base 3. There are no 1s so the next stage is still 0.02020202... This is rewritten as 0.01010101... When read in base 2, this corresponds to 1/3, so ''c''(1/4) = 1/3.',
16 => '* 1/5 is 0.01210121... in base 3. The digits after the first 1 are replaced by 0s to produce 0.01000000... This is not rewritten since there are no 2s. When read in base 2, this corresponds to 1/4, so ''c''(1/5) = 1/4.',
17 => '* 200/243 is 0.21102 (or 0.211012222...) in base 3. The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. When read in base 2, this corresponds to 3/4, so ''c''(200/243) = 3/4.',
18 => '',
19 => 'Equivalently, if <math>\mathcal{C}</math> is the [[Cantor set]] on [0,1], then the Cantor function ''c''<nowiki> : [0,1] → [0,1] can be defined as</nowiki>',
20 => '',
21 => ':<math>c(x) =\begin{cases} ',
22 => ' \sum_{n=1}^\infty \frac{a_n}{2^n}, & x = \sum_{n=1}^\infty ',
23 => '\frac{2a_n}{3^n}\in\mathcal{C}\ \mathrm{for}\ a_n\in\{0,1\};',
24 => '\\ \sup_{y\leq x,\, y\in\mathcal{C}} c(y), & x\in [0,1]\setminus \mathcal{C}.\\ \end{cases}',
25 => '</math>',
26 => '',
27 => 'This formula is well-defined, since every member of the Cantor set has a ''unique'' base 3 representation that only contains the digits 0 or 2. (For some members of <math>\mathcal{C}</math>, the ternary expansion is repeating with trailing 2's and there is an alternative non-repeating expansion ending in 1. For example, 1/3 = 0.1<sub>3</sub> = 0.02222...<sub>3</sub> is a member of the Cantor set). Since ''c''(0) = 0 and ''c''(1) = 1, and ''c'' is monotonic on <math>\mathcal{C}</math>, it is clear that 0 ≤ ''c''(''x'') ≤ 1 also holds for all <math>x\in[0,1]\setminus\mathcal{C}</math>. ',
28 => '',
29 => '==Properties==',
30 => 'The Cantor function challenges naive intuitions about [[continuous function|continuity]] and [[measure (mathematics)|measure]]; though it is continuous everywhere and has zero derivative [[almost everywhere]], <math display="inline">c(x)</math> goes from 0 to 1 as <math display="inline>x</math> goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is [[uniformly continuous]] (precisely, it is [[Hölder continuous]] of exponent ''α'' = log 2/log 3) but not [[absolute continuity|absolutely continuous]]. It is constant on intervals of the form (0.''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>...''x''<sub>n</sub>022222..., 0.''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>...''x''<sub>n</sub>200000...), and every point not in the Cantor set is in one of these intervals, so its derivative is 0 outside of the Cantor set. On the other hand, it has no [[derivative]] at any point in an [[uncountable]] subset of the [[Cantor set]] containing the interval endpoints described above.',
31 => '',
32 => 'The Cantor function can also be seen as the [[cumulative distribution function|cumulative probability distribution function]] of the 1/2-1/2 [[Bernoulli measure]] ''μ'' supported on the Cantor set: <math display="inline">c(x)=\mu([0,x])</math>. This probability distribution, called the [[Cantor distribution]], has no discrete part. That is, the corresponding measure is [[Atom (measure theory)|atomless]]. This is why there are no jump discontinuities in the function; any such jump would correspond to an atom in the measure.',
33 => '',
34 => 'However, no non-constant part of the Cantor function can be represented as an integral of a [[probability density function]]; integrating any putative [[probability density function]] that is not [[almost everywhere]] zero over any interval will give positive probability to some interval to which this distribution assigns probability zero. In particular, as {{harvtxt|Vitali|1905}} pointed out, the function is not the integral of its derivative even though the derivative exists almost everywhere.',
35 => '',
36 => 'The Cantor function is the standard example of a [[singular function]].',
37 => '',
38 => 'The Cantor function is non-decreasing, and so in particular its graph defines a [[rectifiable curve]]. {{harvtxt|Scheeffer|1884}} showed that the arc length of its graph is 2.',
39 => '',
40 => '===Lack of absolute continuity===',
41 => 'Because the [[Lebesgue measure]] of the [[Uncountable set|uncountably infinite]] [[Cantor set]] is 0, for any positive ''ε'' < 1 and ''δ'', there exists a finite sequence of [[pairwise disjoint]] sub-intervals with total length < ''δ'' over which the Cantor function cumulatively rises more than ''ε''.',
42 => '',
43 => 'In fact, for every ''δ'' > 0 there are finitely many pairwise disjoint intervals (''x<SUB>k</SUB>'',''y<SUB>k</SUB>'') (1 ≤ ''k'' ≤ ''M'') with <math>\sum\limits_{k=1}^M (y_k-x_k)<\delta</math> and <math>\sum\limits_{k=1}^M (c(y_k)-c(x_k))=1</math>.',
44 => '',
45 => '== Alternative definitions ==',
46 => '',
47 => '=== Iterative construction ===',
48 => '[[File:Cantor function sequence.png|250px|right]]',
49 => '',
50 => 'Below we define a sequence {''f''<sub>''n''</sub>} of functions on the unit interval that converges to the Cantor function.',
51 => '',
52 => 'Let ''f''<sub>0</sub>(''x'') = ''x''.',
53 => '',
54 => 'Then, for every integer {{nowrap|''n'' ≥ 0}}, the next function ''f''<sub>''n''+1</sub>(''x'') will be defined in terms of ''f''<sub>''n''</sub>(''x'') as follows:',
55 => '',
56 => 'Let ''f''<sub>''n''+1</sub>(''x'') = {{nowrap|1/2 × ''f''<sub>''n''</sub>(3''x'')}}, when {{nowrap|0 ≤ ''x'' ≤ 1/3 }};',
57 => '',
58 => 'Let ''f''<sub>''n''+1</sub>(''x'') = 1/2, when {{nowrap|1/3 ≤ ''x'' ≤ 2/3 }};',
59 => '',
60 => 'Let ''f''<sub>''n''+1</sub>(''x'') = {{nowrap|1/2 + 1/2 × ''f''<sub>''n''</sub>(3 ''x'' − 2)}}, when {{nowrap|2/3 ≤ ''x'' ≤ 1}}.',
61 => '',
62 => 'The three definitions are compatible at the end-points 1/3 and 2/3, because ''f''<sub>''n''</sub>(0) = 0 and ''f''<sub>''n''</sub>(1) = 1 for every ''n'', by induction. One may check that ''f''<sub>''n''</sub> converges pointwise to the Cantor function defined above. Furthermore, the convergence is uniform. Indeed, separating into three cases, according to the definition of ''f''<sub>''n''+1</sub>, one sees that',
63 => '',
64 => ':<math>\max_{x \in [0, 1]} |f_{n+1}(x) - f_n(x)| \le \frac 1 2 \, \max_{x \in [0, 1]} |f_{n}(x) - f_{n-1}(x)|, \quad n \ge 1.</math>',
65 => '',
66 => 'If ''f'' denotes the limit function, it follows that, for every ''n'' ≥ 0,',
67 => '',
68 => ':<math>\max_{x \in [0, 1]} |f(x) - f_n(x)| \le 2^{-n+1} \, \max_{x \in [0, 1]} |f_1(x) - f_0(x)|.</math>',
69 => '',
70 => 'Also the choice of starting function does not really matter, provided ''f''<sub>0</sub>(0) = 0, ''f''<sub>0</sub>(1) = 1 and ''f''<sub>0</sub> is [[Bounded function|bounded]]{{citation needed|date=September 2014}}.',
71 => '',
72 => '=== Fractal volume ===',
73 => 'The Cantor function is closely related to the [[Cantor set]]. The Cantor set ''C'' can be defined as the set of those numbers in the interval [0, 1] that do not contain the digit 1 in their [[base (exponentiation)|base-3 (triadic) expansion]], except if the 1 is followed by zeros only (in which case the tail 1000<math>\ldots</math> can be replaced by 0222<math>\ldots</math> to get rid of any 1). It turns out that the Cantor set is a [[fractal]] with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume). Only the ''D''-dimensional volume <math> H_D </math> (in the sense of a [[Hausdorff dimension|Hausdorff-measure]]) takes a finite value, where <math> D = \log(2)/\log(3) </math> is the fractal dimension of ''C''. We may define the Cantor function alternatively as the ''D''-dimensional volume of sections of the Cantor set'
] |
External links added in the edit (added_links) | [] |
External links removed in the edit (removed_links) | [
0 => 'http://mathworld.wolfram.com/CantorStaircaseFunction.html'
] |
External links in the page, before the edit (old_links) | [
0 => '//arxiv.org/abs/1301.1286',
1 => '//arxiv.org/abs/1301.1286',
2 => '//doi.org/10.1007%2Fbf02418423',
3 => '//doi.org/10.1007%2Fbf02418423',
4 => '//doi.org/10.1007%2Fbf02421552',
5 => '//doi.org/10.1007%2Fbf02421552',
6 => '//doi.org/10.1016%2Fj.exmath.2005.05.002',
7 => '//doi.org/10.1016%2Fj.exmath.2005.05.002',
8 => '//doi.org/10.1017%2FS0305004103006960',
9 => '//doi.org/10.1017%2FS0305004103006960',
10 => '//doi.org/10.1017%2FS0305004113000698',
11 => '//doi.org/10.1017%2FS0305004113000698',
12 => '//doi.org/10.2307%2F2159830',
13 => '//doi.org/10.2307%2F2159830',
14 => '//doi.org/10.2307%2F2690689',
15 => '//doi.org/10.2307%2F2690689',
16 => '//www.ams.org/mathscinet-getitem?mr=2195181',
17 => '//www.ams.org/mathscinet-getitem?mr=2195181',
18 => '//www.ams.org/mathscinet-getitem?mr=2681574',
19 => '//www.ams.org/mathscinet-getitem?mr=2681574',
20 => '//www.jstor.org/stable/2159830',
21 => '//www.jstor.org/stable/2159830',
22 => '//www.jstor.org/stable/2690689',
23 => '//www.jstor.org/stable/2690689',
24 => '//www.worldcat.org/issn/0001-5962',
25 => '//www.worldcat.org/issn/0001-5962',
26 => '//www.worldcat.org/issn/0025-570X',
27 => '//www.worldcat.org/issn/0025-570X',
28 => '//www.worldcat.org/issn/0723-0869',
29 => '//www.worldcat.org/issn/0723-0869',
30 => '//www.worldcat.org/issn/1469-8064',
31 => '//www.worldcat.org/issn/1469-8064',
32 => '//www.worldcat.org/oclc/976406106',
33 => '//www.worldcat.org/oclc/976406106',
34 => 'http://bookstore.ams.org/gsm-181/',
35 => 'http://demonstrations.wolfram.com/CantorFunction/',
36 => 'http://mathworld.wolfram.com/CantorStaircaseFunction.html',
37 => 'http://users.utu.fi/vuorinen/REA12/107.pdf',
38 => 'https://api.semanticscholar.org/CorpusID:56402751',
39 => 'https://mathworld.wolfram.com/CantorFunction.html',
40 => 'https://ui.adsabs.harvard.edu/abs/2004MPCPS.136..167F',
41 => 'https://ui.adsabs.harvard.edu/abs/2014MPCPS.156..295T',
42 => 'https://www.encyclopediaofmath.org/index.php/Cantor_ternary_function'
] |
Parsed HTML source of the new revision (new_html) | '<div class="mw-parser-output"><p class="mw-empty-elt">
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<div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Continuous function that is not absolutely continuous</div>
<div class="thumb tright"><div class="thumbinner" style="width:402px;"><a href="/wiki/File:CantorEscalier.svg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/CantorEscalier.svg/400px-CantorEscalier.svg.png" decoding="async" width="400" height="400" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/CantorEscalier.svg/600px-CantorEscalier.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/CantorEscalier.svg/800px-CantorEscalier.svg.png 2x" data-file-width="700" data-file-height="700" /></a> <div class="thumbcaption"><div class="magnify"><a href="/wiki/File:CantorEscalier.svg" class="internal" title="Enlarge"></a></div>An approximation to the graph of the Cantor function on the <a href="/wiki/Unit_interval" title="Unit interval">unit interval</a></div></div></div>
<p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>Cantor function</b> is an example of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> that is <a href="/wiki/Continuous_function" title="Continuous function">continuous</a>, but not <a href="/wiki/Absolute_continuity" title="Absolute continuity">absolutely continuous</a>. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reachmath> can be replaced by 0222<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ldots }">
<semantics>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8619532e44ee1ccae3ab03405a6885260d09ed" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="\ldots "/></span> to get rid of any 1). It turns out that the Cantor set is a <a href="/wiki/Fractal" title="Fractal">fractal</a> with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume). Only the <i>D</i>-dimensional volume <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{D}}">
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<annotation encoding="application/x-tex">{\displaystyle H_{D}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e58cab9170393fc1dcec5229a0b1c1420ebfee82" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.524ex; height:2.509ex;" alt=" H_D "/></span> (in the sense of a <a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff-measure</a>) takes a finite value, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=\log(2)/\log(3)}">
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<annotation encoding="application/x-tex">{\displaystyle D=\log(2)/\log(3)}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bbcd2b71b264491d79963c96b129ce878be34ba" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:18.459ex; height:2.843ex;" alt=" D = \log(2)/\log(3) "/></span> is the fractal dimension of <i>C</i>. We may define the Cantor function alternatively as the <i>D</i>-dimensional volume of sections of the Cantor set
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=H_{D}(C\cap (0,x)).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle f(x)=H_{D}(C\cap (0,x)).}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f66653840e347772c181fa0e506ee4b1d57133e7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:23.181ex; height:2.843ex;" alt=" f(x)=H_D(C \cap (0,x)). "/></span></dd></dl>
<div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Self-similarity"><span class="tocnumber">1</span> <span class="toctext">Self-similarity</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Generalizations"><span class="tocnumber">2</span> <span class="toctext">Generalizations</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#See_also"><span class="tocnumber">3</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="#Notes"><span class="tocnumber">4</span> <span class="toctext">Notes</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#References"><span class="tocnumber">5</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-6"><a href="#External_links"><span class="tocnumber">6</span> <span class="toctext">External links</span></a></li>
</ul>
</div>
<h2><span class="mw-headline" id="Self-similarity">Self-similarity</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor_function&action=edit&section=1" title="Edit section: Self-similarity">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>The Cantor function possesses several <a href="/wiki/Symmetry" title="Symmetry">symmetries</a>. For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x\leq 1}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle 0\leq x\leq 1}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30810e06ad49f3a837bd2193d4392eda1f74e7ab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:9.852ex; height:2.343ex;" alt="0\leq x\leq 1"/></span>, there is a reflection symmetry
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(x)=1-c(1-x)}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle c(x)=1-c(1-x)}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50c4630c66cd065f55c1bdad2670f3c4d1cec7c4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:19.396ex; height:2.843ex;" alt="{\displaystyle c(x)=1-c(1-x)}"/></span></dd></dl>
<p>and a pair of magnifications, one on the left and one on the right:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\left({\frac {x}{3}}\right)={\frac {c(x)}{2}}}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle c\left({\frac {x}{3}}\right)={\frac {c(x)}{2}}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1afdb66829128e8c0cf969edb860bf7ba6ae35" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:14.416ex; height:5.676ex;" alt="{\displaystyle c\left({\frac {x}{3}}\right)={\frac {c(x)}{2}}}"/></span></dd></dl>
<p>and
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\left({\frac {x+2}{3}}\right)={\frac {1+c(x)}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle c\left({\frac {x+2}{3}}\right)={\frac {1+c(x)}{2}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32379b4899e7d67d361395c099723f2fb0b76dbb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:23.067ex; height:6.343ex;" alt="{\displaystyle c\left({\frac {x+2}{3}}\right)={\frac {1+c(x)}{2}}}"/></span></dd></dl>
<p>The magnifications can be cascaded; they generate the <a href="/wiki/Dyadic_monoid" class="mw-redirect" title="Dyadic monoid">dyadic monoid</a>. This is exhibited by defining several helper functions. Define the reflection as
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(x)=1-x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo stretchy="false">(</mo>
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<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r(x)=1-x}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e90cbdd7ef4a309d85dcab04d97d49cffd006e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:12.619ex; height:2.843ex;" alt="{\displaystyle r(x)=1-x}"/></span></dd></dl>
<p>The first self-symmetry can be expressed as
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\circ c=c\circ r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo>∘<!-- ∘ --></mo>
<mi>c</mi>
<mo>=</mo>
<mi>c</mi>
<mo>∘<!-- ∘ --></mo>
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r\circ c=c\circ r}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83dcb939440bbe714db76d6ed86060aad90e2b4c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:11.599ex; height:1.676ex;" alt="{\displaystyle r\circ c=c\circ r}"/></span></dd></dl>
<p>where the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∘<!-- ∘ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \circ }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="\circ "/></span> denotes function composition. That is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r\circ c)(x)=r(c(x))=1-c(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo>∘<!-- ∘ --></mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>r</mi>
<mo stretchy="false">(</mo>
<mi>c</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (r\circ c)(x)=r(c(x))=1-c(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dadd68341b89477c45359b3c17be6178059332f5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:30.548ex; height:2.843ex;" alt="{\displaystyle (r\circ c)(x)=r(c(x))=1-c(x)}"/></span> and likewise for the other cases. For the left and right magnifications, write the left-mappings
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{D}(x)={\frac {x}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L_{D}(x)={\frac {x}{2}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a66038b13ecc79d3ba113ecc9466c7e23cb5b4ed" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:11.579ex; height:4.676ex;" alt="{\displaystyle L_{D}(x)={\frac {x}{2}}}"/></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{C}(x)={\frac {x}{3}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>C</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>3</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L_{C}(x)={\frac {x}{3}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be34577e4f8addc545f15fc87a3885e031f2314e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:11.467ex; height:4.676ex;" alt="{\displaystyle L_{C}(x)={\frac {x}{3}}}"/></span></dd></dl>
<p>Then the Cantor function obeys
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{D}\circ c=c\circ L_{C}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
<mo>∘<!-- ∘ --></mo>
<mi>c</mi>
<mo>=</mo>
<mi>c</mi>
<mo>∘<!-- ∘ --></mo>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>C</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L_{D}\circ c=c\circ L_{C}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaa3a187f8db401f91be33cc4fbd3c6bc1f9abe3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:15.741ex; height:2.509ex;" alt="{\displaystyle L_{D}\circ c=c\circ L_{C}}"/></span></dd></dl>
<p>Similarly, define the right mappings as
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{D}(x)={\frac {1+x}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{D}(x)={\frac {1+x}{2}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d17b46c69f2dd7681ee239763b68687bd4d608ea" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:15.763ex; height:5.176ex;" alt="{\displaystyle R_{D}(x)={\frac {1+x}{2}}}"/></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{C}(x)={\frac {2+x}{3}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>C</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{C}(x)={\frac {2+x}{3}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f533a0921725d3c42c03cb9bd3d084f7d1722f5e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:15.651ex; height:5.176ex;" alt="{\displaystyle R_{C}(x)={\frac {2+x}{3}}}"/></span></dd></dl>
<p>Then, likewise,
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{D}\circ c=c\circ R_{C}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
<mo>∘<!-- ∘ --></mo>
<mi>c</mi>
<mo>=</mo>
<mi>c</mi>
<mo>∘<!-- ∘ --></mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>C</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{D}\circ c=c\circ R_{C}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8b381a11120c1ef99778926a8f46c56a7e1690" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:16.104ex; height:2.509ex;" alt="{\displaystyle R_{D}\circ c=c\circ R_{C}}"/></span></dd></dl>
<p>The two sides can be mirrored one onto the other, in that
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{D}\circ r=r\circ R_{D}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
<mo>∘<!-- ∘ --></mo>
<mi>r</mi>
<mo>=</mo>
<mi>r</mi>
<mo>∘<!-- ∘ --></mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L_{D}\circ r=r\circ R_{D}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195da20eac1c8b064af3cc2766c76f06eee61083" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:16.118ex; height:2.509ex;" alt="{\displaystyle L_{D}\circ r=r\circ R_{D}}"/></span></dd></dl>
<p>and likewise,
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{C}\circ r=r\circ R_{C}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>C</mi>
</mrow>
</msub>
<mo>∘<!-- ∘ --></mo>
<mi>r</mi>
<mo>=</mo>
<mi>r</mi>
<mo>∘<!-- ∘ --></mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>C</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L_{C}\circ r=r\circ R_{C}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89e22962f25fcbf0ce2d85217ab1682bbf528cf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:15.895ex; height:2.509ex;" alt="{\displaystyle L_{C}\circ r=r\circ R_{C}}"/></span></dd></dl>
<p>These operations can be stacked arbitrarily. Consider, for example, the sequence of left-right moves <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle LRLLR.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mi>R</mi>
<mi>L</mi>
<mi>L</mi>
<mi>R</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle LRLLR.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd9a7e192e435f0622d170c44cb9d4cf4f677271" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:8.923ex; height:2.176ex;" alt="{\displaystyle LRLLR.}"/></span> Adding the subscripts C and D, and, for clarity, dropping the composition operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∘<!-- ∘ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \circ }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="\circ "/></span> in all but a few places, one has:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{D}R_{D}L_{D}L_{D}R_{D}\circ c=c\circ L_{C}R_{C}L_{C}L_{C}R_{C}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
<mo>∘<!-- ∘ --></mo>
<mi>c</mi>
<mo>=</mo>
<mi>c</mi>
<mo>∘<!-- ∘ --></mo>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>C</mi>
</mrow>
</msub>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>C</mi>
</mrow>
</msub>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>C</mi>
</mrow>
</msub>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>C</mi>
</mrow>
</msub>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>C</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L_{D}R_{D}L_{D}L_{D}R_{D}\circ c=c\circ L_{C}R_{C}L_{C}L_{C}R_{C}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf9f88adc7ad4ace496598979fe22f707f471b4f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:41.425ex; height:2.509ex;" alt="{\displaystyle L_{D}R_{D}L_{D}L_{D}R_{D}\circ c=c\circ L_{C}R_{C}L_{C}L_{C}R_{C}}"/></span></dd></dl>
<p>Arbitrary finite-length strings in the letters L and R correspond to the <a href="/wiki/Dyadic_rationals" class="mw-redirect" title="Dyadic rationals">dyadic rationals</a>, in that every dyadic rational can be written as both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=n/2^{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y=n/2^{m}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b93dfeeadce8a57f173d5e15fc970b513d362ff9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.649ex; height:2.843ex;" alt="{\displaystyle y=n/2^{m}}"/></span> for integer <i>n</i> and <i>m</i> and as finite length of bits <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=0.b_{1}b_{2}b_{3}\cdots b_{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<mn>0.</mn>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>⋯<!-- ⋯ --></mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y=0.b_{1}b_{2}b_{3}\cdots b_{m}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a04a56b38079089ee5cff10482be90a326869cfd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:18.389ex; height:2.509ex;" alt="{\displaystyle y=0.b_{1}b_{2}b_{3}\cdots b_{m}}"/></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{k}\in \{0,1\}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b_{k}\in \{0,1\}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a216bac798702ebe1a2dba84c088ca2a2dd6128" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.258ex; height:2.843ex;" alt="{\displaystyle b_{k}\in \{0,1\}.}"/></span> Thus, every dyadic rational is in one-to-one correspondence with some self-symmetry of the Cantor function.
</p><p>Some notational rearrangements can make the above slightly easier to express. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d13273b9af4564fa2c421c96d039c414db8628" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="g_{0}"/></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3755e3e04ec295992b2b5331655ef83a500a05c1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="g_{1}"/></span> stand for L and R. Function composition extends this to a <a href="/wiki/Monoid" title="Monoid">monoid</a>, in that one can write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{010}=g_{0}g_{1}g_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>010</mn>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{010}=g_{0}g_{1}g_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6e259c8a6d532d13783e5d6f423ad022b1738c0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:13.396ex; height:2.009ex;" alt="{\displaystyle g_{010}=g_{0}g_{1}g_{0}}"/></span> and generally, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{A}g_{B}=g_{AB}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
</mrow>
</msub>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>B</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>A</mi>
<mi>B</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{A}g_{B}=g_{AB}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f75be0a170d2924de25755e6132645c6d48e07d8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:12.082ex; height:2.009ex;" alt="{\displaystyle g_{A}g_{B}=g_{AB}}"/></span> for some binary strings of digits <i>A</i>, <i>B</i>, where <i>AB</i> is just the ordinary <a href="/wiki/Concatenation" title="Concatenation">concatenation</a> of such strings. The dyadic monoid <i>M</i> is then the monoid of all such finite-length left-right moves. Writing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \in M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>γ<!-- γ --></mi>
<mo>∈<!-- ∈ --></mo>
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \gamma \in M}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e4c574bcce05b99e8f917a3c0d50c9ca733922" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.545ex; height:2.676ex;" alt="{\displaystyle \gamma \in M}"/></span> as a general element of the monoid, there is a corresponding self-symmetry of the Cantor function:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{D}\circ c=c\circ \gamma _{C}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>γ<!-- γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
<mo>∘<!-- ∘ --></mo>
<mi>c</mi>
<mo>=</mo>
<mi>c</mi>
<mo>∘<!-- ∘ --></mo>
<msub>
<mi>γ<!-- γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>C</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \gamma _{D}\circ c=c\circ \gamma _{C}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0690c5bcf10fdbdb6815ba24e36c24e7047fedab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.984ex; height:2.176ex;" alt="{\displaystyle \gamma _{D}\circ c=c\circ \gamma _{C}}"/></span></dd></dl>
<p>The dyadic monoid itself has several interesting properties. It can be viewed as a finite number of left-right moves down an infinite <a href="/wiki/Binary_tree" title="Binary tree">binary tree</a>; the infinitely distant "leaves" on the tree correspond to the points on the Cantor set, and so, the monoid also represents the self-symmetries of the Cantor set. In fact, a large class of commonly occurring fractals are described by the dyadic monoid; additional examples can be found in the article on <a href="/wiki/De_Rham_curve" title="De Rham curve">de Rham curves</a>. Other fractals possessing self-similarity are described with other kinds of monoids. The dyadic monoid is itself a sub-monoid of the <a href="/wiki/Modular_group" title="Modular group">modular group</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SL(2,\mathbb {Z} ).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mi>L</mi>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle SL(2,\mathbb {Z} ).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f38ac4cc941b61061f523f1cbee989dac976be3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.285ex; height:2.843ex;" alt="{\displaystyle SL(2,\mathbb {Z} ).}"/></span>
</p><p>Note that the Cantor function bears more than a passing resemblance to <a href="/wiki/Minkowski%27s_question-mark_function" title="Minkowski's question-mark function">Minkowski's question-mark function</a>. In particular, it obeys the exact same symmetry relations, although in an altered form.
</p>
<h2><span class="mw-headline" id="Generalizations">Generalizations</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor_function&action=edit&section=2" title="Edit section: Generalizations">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>Let
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\sum _{k=1}^{\infty }b_{k}2^{-k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>k</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y=\sum _{k=1}^{\infty }b_{k}2^{-k}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55fe50ed339e1655e25f2d970da76c13075ea1f1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:13.612ex; height:6.843ex;" alt="y=\sum_{k=1}^\infty b_k 2^{-k}"/></span></dd></dl>
<p>be the <a href="/wiki/Dyadic_rational" title="Dyadic rational">dyadic</a> (binary) expansion of the real number 0 ≤ <i>y</i> ≤ 1 in terms of binary digits <i>b</i><sub><i>k</i></sub> ∈ {0,1}. This expansion is discussed in greater detail in the article on the <a href="/wiki/Dyadic_transformation" title="Dyadic transformation">dyadic transformation</a>. Then consider the function
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{z}(y)=\sum _{k=1}^{\infty }b_{k}z^{k}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>C</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C_{z}(y)=\sum _{k=1}^{\infty }b_{k}z^{k}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e70ba22bf66d1fcd00473e4f5a1b16d0879e64d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:17.381ex; height:6.843ex;" alt="C_z(y)=\sum_{k=1}^\infty b_k z^{k}."/></span></dd></dl>
<p>For <i>z</i> = 1/3, the inverse of the function <i>x</i> = 2 <i>C</i><sub>1/3</sub>(<i>y</i>) is the Cantor function. That is, <i>y</i> = <i>y</i>(<i>x</i>) is the Cantor function. In general, for any <i>z</i> < 1/2, <i>C</i><sub><i>z</i></sub>(<i>y</i>) looks like the Cantor function turned on its side, with the width of the steps getting wider as <i>z</i> approaches zero.
</p><p>As mentioned above, the Cantor function is also the cumulative distribution function of a measure on the Cantor set. Different Cantor functions, or Devil's Staircases, can be obtained by considering different atom-less probability measures supported on the Cantor set or other fractals. While the Cantor function has derivative 0 almost everywhere, current research focusses on the question of the size of the set of points where the upper right derivative is distinct from the lower right derivative, causing the derivative to not exist. This analysis of differentiability is usually given in terms of <a href="/wiki/Fractal_dimension" title="Fractal dimension">fractal dimension</a>, with the Hausdorff dimension the most popular choice. This line of research was started in the 1990s by Darst,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup> who showed that the Hausdorff dimension of the set of non-differentiability of the Cantor function is the square of the dimension of the Cantor set, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\log 2/\log 3)^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>log</mi>
<mo>⁡<!-- --></mo>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>log</mi>
<mo>⁡<!-- --></mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\log 2/\log 3)^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db44bc29973f183d97b16b68910c00775937d0ec" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:13.456ex; height:3.176ex;" alt="{\displaystyle (\log 2/\log 3)^{2}}"/></span>. Subsequently <a href="/wiki/Kenneth_Falconer_(mathematician)" title="Kenneth Falconer (mathematician)">Falconer</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup> showed that this squaring relationship holds for all Ahlfor's regular, singular measures, i.e.<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim _{H}\left\{x:f'(x)=\lim _{h\to 0^{+}}{\frac {\mu ([x,x+h])}{h}}{\text{ does not exist}}\right\}=\left(\dim _{H}\operatorname {supp} (\mu )\right)^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>dim</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>H</mi>
</mrow>
</msub>
<mo>⁡<!-- --></mo>
<mrow>
<mo>{</mo>
<mrow>
<mi>x</mi>
<mo>:</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>h</mi>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mn>0</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">(</mo>
<mo stretchy="false">[</mo>
<mi>x</mi>
<mo>,</mo>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">)</mo>
</mrow>
<mi>h</mi>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext> does not exist</mtext>
</mrow>
</mrow>
<mo>}</mo>
</mrow>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>dim</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>H</mi>
</mrow>
</msub>
<mo>⁡<!-- --></mo>
<mi>supp</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \dim _{H}\left\{x:f'(x)=\lim _{h\to 0^{+}}{\frac {\mu ([x,x+h])}{h}}{\text{ does not exist}}\right\}=\left(\dim _{H}\operatorname {supp} (\mu )\right)^{2}}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f78e378bc322fe3f79eacbfe733c18b4a9f0284" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -2.505ex; width:72.931ex; height:6.343ex;" alt="{\displaystyle \dim _{H}\left\{x:f'(x)=\lim _{h\to 0^{+}}{\frac {\mu ([x,x+h])}{h}}{\text{ does not exist}}\right\}=\left(\dim _{H}\operatorname {supp} (\mu )\right)^{2}}"/></div>Later, Troscheit<sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup> obtain a more comprehensive picture of the set where the derivative does not exist for more general normalized Gibb's measures supported on self-conformal and <a href="/wiki/Self-similarity" title="Self-similarity">self-similar sets</a>.
</p><p><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a>'s <a href="/wiki/Minkowski%27s_question_mark_function" class="mw-redirect" title="Minkowski's question mark function">question mark function</a> loosely resembles the Cantor function visually, appearing as a "smoothed out" form of the latter; it can be constructed by passing from a continued fraction expansion to a binary expansion, just as the Cantor function can be constructed by passing from a ternary expansion to a binary expansion. The question mark function has the interesting property of having vanishing derivatives at all rational numbers.
</p>
<h2><span class="mw-headline" id="See_also">See also</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor_function&action=edit&section=3" title="Edit section: See also">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li><a href="/wiki/Dyadic_transformation" title="Dyadic transformation">Dyadic transformation</a></li></ul>
<h2><span class="mw-headline" id="Notes">Notes</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor_function&action=edit&section=4" title="Edit section: Notes">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r999302996">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFDarst1993" class="citation journal cs1">Darst, Richard (1993-09-01). "The Hausdorff Dimension of the Nondifferentiability Set of the Cantor Function is [ ln(2)/ln(3) ]2". <i>Proceedings of the American Mathematical Society</i>. <b>119</b> (1): 105–108. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2159830">10.2307/2159830</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="//www.jstor.org/stable/2159830">2159830</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=The+Hausdorff+Dimension+of+the+Nondifferentiability+Set+of+the+Cantor+Function+is+%5B+ln%282%29%2Fln%283%29+%5D2&rft.volume=119&rft.issue=1&rft.pages=105-108&rft.date=1993-09-01&rft_id=info%3Adoi%2F10.2307%2F2159830&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2159830%23id-name%3DJSTOR&rft.aulast=Darst&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFFalconer2004" class="citation journal cs1">Falconer, Kenneth J. (2004-01-01). "One-sided multifractal analysis and points of non-differentiability of devil's staircases". <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. <b>136</b> (1): 167–174. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004MPCPS.136..167F">2004MPCPS.136..167F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0305004103006960">10.1017/S0305004103006960</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/1469-8064">1469-8064</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Proceedings+of+the+Cambridge+Philosophical+Society&rft.atitle=One-sided+multifractal+analysis+and+points+of+non-differentiability+of+devil%27s+staircases&rft.volume=136&rft.issue=1&rft.pages=167-174&rft.date=2004-01-01&rft.issn=1469-8064&rft_id=info%3Adoi%2F10.1017%2FS0305004103006960&rft_id=info%3Abibcode%2F2004MPCPS.136..167F&rft.aulast=Falconer&rft.aufirst=Kenneth+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></span>
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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFTroscheit2014" class="citation journal cs1">Troscheit, Sascha (2014-03-01). "Hölder differentiability of self-conformal devil's staircases". <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. <b>156</b> (2): 295–311. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="//arxiv.org/abs/1301.1286">1301.1286</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014MPCPS.156..295T">2014MPCPS.156..295T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0305004113000698">10.1017/S0305004113000698</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/1469-8064">1469-8064</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:56402751">56402751</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Proceedings+of+the+Cambridge+Philosophical+Society&rft.atitle=H%C3%B6lder+differentiability+of+self-conformal+devil%27s+staircases&rft.volume=156&rft.issue=2&rft.pages=295-311&rft.date=2014-03-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A56402751%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2014MPCPS.156..295T&rft_id=info%3Aarxiv%2F1301.1286&rft.issn=1469-8064&rft_id=info%3Adoi%2F10.1017%2FS0305004113000698&rft.aulast=Troscheit&rft.aufirst=Sascha&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></span>
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</ol></div>
<h2><span class="mw-headline" id="References">References</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor_function&action=edit&section=5" title="Edit section: References">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFBass2013" class="citation book cs1"><a href="/wiki/Richard_F._Bass" title="Richard F. Bass">Bass, Richard Franklin</a> (2013) [2011]. <i>Real analysis for graduate students</i> (Second ed.). Createspace Independent Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4818-6914-0" title="Special:BookSources/978-1-4818-6914-0"><bdi>978-1-4818-6914-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+analysis+for+graduate+students&rft.edition=Second&rft.pub=Createspace+Independent+Publishing&rft.date=2013&rft.isbn=978-1-4818-6914-0&rft.aulast=Bass&rft.aufirst=Richard+Franklin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFCantor1884" class="citation journal cs1">Cantor, G. (1884). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02418423">"De la puissance des ensembles parfaits de points: Extrait d'une lettre adressée à l'éditeur"</a> [The power of perfect sets of points: Extract from a letter addressed to the editor]. <i>Acta Mathematica</i>. International Press of Boston. <b>4</b>: 381–392. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02418423">10.1007/bf02418423</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0001-5962">0001-5962</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Mathematica&rft.atitle=De+la+puissance+des+ensembles+parfaits+de+points%3A+Extrait+d%27une+lettre+adress%C3%A9e+%C3%A0+l%27%C3%A9diteur&rft.volume=4&rft.pages=381-392&rft.date=1884&rft_id=info%3Adoi%2F10.1007%2Fbf02418423&rft.issn=0001-5962&rft.aulast=Cantor&rft.aufirst=G.&rft_id=%2F%2Fdoi.org%2F10.1007%252Fbf02418423&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span> Reprinted in: E. Zermelo (Ed.), Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts, Springer, New York, 1980.</li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFDarstPalagalloPrice2010" class="citation cs2">Darst, Richard B.; Palagallo, Judith A.; Price, Thomas E. (2010), <i>Curious curves</i>, Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4291-28-6" title="Special:BookSources/978-981-4291-28-6"><bdi>978-981-4291-28-6</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=2681574">2681574</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Curious+curves&rft.place=Hackensack%2C+NJ&rft.pub=World+Scientific+Publishing+Co.+Pte.+Ltd.&rft.date=2010&rft.isbn=978-981-4291-28-6&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D2681574%23id-name%3DMR&rft.aulast=Darst&rft.aufirst=Richard+B.&rft.au=Palagallo%2C+Judith+A.&rft.au=Price%2C+Thomas+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFDovgosheyMartioRyazanovVuorinen2006" class="citation journal cs1">Dovgoshey, O.; Martio, O.; Ryazanov, V.; Vuorinen, M. (2006). <a rel="nofollow" class="external text" href="http://users.utu.fi/vuorinen/REA12/107.pdf">"The Cantor function"</a> <span class="cs1-format">(PDF)</span>. <i>Expositiones Mathematicae</i>. Elsevier BV. <b>24</b> (1): 1–37. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.exmath.2005.05.002">10.1016/j.exmath.2005.05.002</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0723-0869">0723-0869</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=2195181">2195181</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Expositiones+Mathematicae&rft.atitle=The+Cantor+function&rft.volume=24&rft.issue=1&rft.pages=1-37&rft.date=2006&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D2195181%23id-name%3DMR&rft.issn=0723-0869&rft_id=info%3Adoi%2F10.1016%2Fj.exmath.2005.05.002&rft.aulast=Dovgoshey&rft.aufirst=O.&rft.au=Martio%2C+O.&rft.au=Ryazanov%2C+V.&rft.au=Vuorinen%2C+M.&rft_id=http%3A%2F%2Fusers.utu.fi%2Fvuorinen%2FREA12%2F107.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFFleron1994" class="citation journal cs1">Fleron, Julian F. (1994-04-01). "A Note on the History of the Cantor Set and Cantor Function". <i>Mathematics Magazine</i>. Informa UK Limited. <b>67</b> (2): 136–140. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2690689">10.2307/2690689</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0025-570X">0025-570X</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="//www.jstor.org/stable/2690689">2690689</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=A+Note+on+the+History+of+the+Cantor+Set+and+Cantor+Function&rft.volume=67&rft.issue=2&rft.pages=136-140&rft.date=1994-04-01&rft.issn=0025-570X&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2690689%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2690689&rft.aulast=Fleron&rft.aufirst=Julian+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFLebesgue1904" class="citation cs2">Lebesgue, H. (1904), <i>Leçons sur l'intégration et la recherche des fonctions primitives</i> [<i>Lessons on integration and search for primitive functions</i>], Paris: Gauthier-Villars</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Le%C3%A7ons+sur+l%27int%C3%A9gration+et+la+recherche+des+fonctions+primitives&rft.place=Paris&rft.pub=Gauthier-Villars&rft.date=1904&rft.aulast=Lebesgue&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFLeoni2017" class="citation book cs1">Leoni, Giovanni (2017). <a rel="nofollow" class="external text" href="http://bookstore.ams.org/gsm-181/"><i>A first course in Sobolev spaces</i></a>. <b>181</b> (2nd ed.). Providence, Rhode Island: American Mathematical Society. p. 734. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-2921-8" title="Special:BookSources/978-1-4704-2921-8"><bdi>978-1-4704-2921-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/oclc/976406106">976406106</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+first+course+in+Sobolev+spaces&rft.place=Providence%2C+Rhode+Island&rft.pages=734&rft.edition=2nd&rft.pub=American+Mathematical+Society&rft.date=2017&rft_id=info%3Aoclcnum%2F976406106&rft.isbn=978-1-4704-2921-8&rft.aulast=Leoni&rft.aufirst=Giovanni&rft_id=http%3A%2F%2Fbookstore.ams.org%2Fgsm-181%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFScheeffer1884" class="citation journal cs1">Scheeffer, Ludwig (1884). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02421552">"Allgemeine Untersuchungen über Rectification der Curven"</a> [General investigations on rectification of the curves]. <i>Acta Mathematica</i>. International Press of Boston. <b>5</b>: 49–82. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02421552">10.1007/bf02421552</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0001-5962">0001-5962</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Mathematica&rft.atitle=Allgemeine+Untersuchungen+%C3%BCber+Rectification+der+Curven&rft.volume=5&rft.pages=49-82&rft.date=1884&rft_id=info%3Adoi%2F10.1007%2Fbf02421552&rft.issn=0001-5962&rft.aulast=Scheeffer&rft.aufirst=Ludwig&rft_id=%2F%2Fdoi.org%2F10.1007%252Fbf02421552&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFThomsonBrucknerBruckner2008" class="citation book cs1">Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008) [2001]. <i>Elementary real analysis</i> (Second ed.). ClassicalRealAnalysis.com. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4348-4367-8" title="Special:BookSources/978-1-4348-4367-8"><bdi>978-1-4348-4367-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+real+analysis&rft.edition=Second&rft.pub=ClassicalRealAnalysis.com&rft.date=2008&rft.isbn=978-1-4348-4367-8&rft.aulast=Thomson&rft.aufirst=Brian+S.&rft.au=Bruckner%2C+Judith+B.&rft.au=Bruckner%2C+Andrew+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFVestrup2003" class="citation book cs1">Vestrup, E.M. (2003). <i>The theory of measures and integration</i>. Wiley series in probability and statistics. John Wiley & sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0471249771" title="Special:BookSources/978-0471249771"><bdi>978-0471249771</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+theory+of+measures+and+integration&rft.series=Wiley+series+in+probability+and+statistics&rft.pub=John+Wiley+%26+sons&rft.date=2003&rft.isbn=978-0471249771&rft.aulast=Vestrup&rft.aufirst=E.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFVitali1905" class="citation cs2">Vitali, A. (1905), "Sulle funzioni integrali" [On the integral functions], <i>Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.</i>, <b>40</b>: 1021–1034</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Atti+Accad.+Sci.+Torino+Cl.+Sci.+Fis.+Mat.+Natur.&rft.atitle=Sulle+funzioni+integrali&rft.volume=40&rft.pages=1021-1034&rft.date=1905&rft.aulast=Vitali&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></li></ul>
<h2><span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor_function&action=edit&section=6" title="Edit section: External links">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Cantor_ternary_function"><i>Cantor ternary function</i> at Encyclopaedia of Mathematics</a></li>
<li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/CantorFunction/">Cantor Function</a> by Douglas Rivers, the <a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>.</li>
<li><span class="citation mathworld" id="Reference-Mathworld-Cantor_Function"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r999302996"/><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CantorFunction.html">"Cantor Function"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Cantor+Function&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCantorFunction.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor+function" class="Z3988"></span></span></li></ul>
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