Cantor function: Difference between revisions

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{{Short description|Continuous function that is not absolutely continuous}}

[[File:CantorEscalier-2.svg|thumb|right|400px|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 [[Pathological_(mathematics)#Pathological_example|counterexample]] in analysis, because it challenges naive intuitions about continuity, [[derivative]], and [[Measure (mathematics)|measure]]. ThoughAlthough it is continuous everywhere, and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reachesgoes from 0 to 1. Thus, in one sensewhile the function seems very much like a constant one whichthat cannot grow, and in another, it does indeed [[Monotonic function|monotonically]] grow.

It is also called 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>{{Cite web|url=http://mathworld.wolfram.com/CantorStaircaseFunction.html|title=Cantor Staircase Function}}</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==

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Equivalently, if <math>\mathcal{C}</math> is the [[Cantor set]] on [0,1], then the Cantor function <math>c:[0,1]\to[0,1]</math> can be defined as

:<math display="block">c(x) =\begin{cases}

\displaystyle \sum_{n=1}^\infty \frac{a_n}{2^n}, & x = \sum_{n=1}^\infty

& \displaystyle \text{if } x = \sum_{n=1}^\infty

\frac{2a_n}{3^n}\in\mathcal{C}\ \mathrm{for}\ a_n\in\{0,1\};

\\ \sup_frac{y\leq x,\, y2a_n}{3^n}\in\mathcal{C}\ \text{for}\ c(y), & xa_n\in [\{0,1]\smallsetminus \mathcal{C}. \end{cases};

\\

\displaystyle \sup_{y\leq x,\, y\in\mathcal{C}} c(y),

& \displaystyle \text{if } x\in [0,1] \setminus \mathcal{C}.

\end{cases}

</math>

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==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 <math>\alpha ''α''&nbsp;=&nbsp;log&nbsp; \log_3(2)</log&nbsp;3math>) but not [[absolute continuity|absolutely continuous]]. It is constant on intervals of the form (0.''x''₁''x''₂''x''₃...''x''_n022222..., 0.''x''₁''x''₂''x''₃...''x''_n200000...), 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.

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===Lack of absolute continuity===

Because theThe [[Lebesgue measure]] of the [[Uncountable set|uncountably infinite]] [[Cantor set]] is 0. Therefore, for any positive ''ε''&nbsp;<&nbsp;1 and any ''δ'' > 0, there exists a finite sequence of [[pairwise disjoint]] sub-intervals with total length <&nbsp;''δ'' over which the Cantor function cumulatively rises more than&nbsp;''ε''.

In fact, for every ''δ''&nbsp;>&nbsp;0 there are finitely many pairwise disjoint intervals (''x<SUB>k</SUB>'',''y<SUB>k</SUB>'') (1&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;''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>.

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[[File:Cantor function sequence.png|250px|right]]

Below we define a sequence {''f''<submath>''n''(f_n)_n</submath>} of functions on the unit interval that converges to the Cantor function.

Let ''f''<sub>0</submath>f_0(''x'') = ''x''</math>.

Then, for every integer <math>n \geq 0</math>, the next function <math>f_{n + 1}(x)</math> will be defined in terms of <math>f_n(x)</math> as follows:<math display="block">f_{n + 1}(x) = \begin{cases} \displaystyle \frac{1}{2} f_n(3 x) &\text{if } 0 \leq x \leq \frac{1}{3} \\ \displaystyle \frac{1}{2} &\text{if } \frac{1}{3} \leq x \leq \frac{2}{3} \\ \displaystyle \frac{1}{2} + \frac{1}{2} f_n(3 x - 2) &\text{if } \frac{2}{3} \leq x \leq 1 \end{cases}</math>The three definitions are compatible at the end-points <math>\tfrac{1}{3}</math> and <math>\tfrac{2}{3}</math>, because <math>f_n(0) = 0</math> and <math>f_n(1) = 1</math> for every <math>n</math>, by induction. One may check that <math>(f_n)_n</math> converges pointwise to the Cantor function defined above. Furthermore, the convergence is uniform. Indeed, separating into three cases, according to the definition of <math>f_{n + 1}</math>, one sees that

Then, for every integer {{nowrap|''n'' &ge; 0}}, the next function ''f''_''n''+1(''x'') will be defined in terms of ''f''_''n''(''x'') as follows:

Let ''f''_''n''+1(''x'')&nbsp;= {{nowrap|1/2 &times; ''f''_''n''(3''x'')}},&nbsp; when {{nowrap|0 ≤ ''x'' ≤ 1/3&thinsp;}};

Let ''f''_''n''+1(''x'')&nbsp;= 1/2,&nbsp; when {{nowrap|1/3 ≤ ''x'' ≤ 2/3&thinsp;}};

Let ''f''_''n''+1(''x'')&nbsp;= {{nowrap|1/2 + 1/2 &times; ''f''_''n''(3&thinsp;''x'' &minus; 2)}},&nbsp; when {{nowrap|2/3 ≤ ''x'' ≤ 1}}.

The three definitions are compatible at the end-points 1/3 and 2/3, because ''f''_''n''(0)&nbsp;= 0 and ''f''_''n''(1)&nbsp;= 1 for every&nbsp;''n'', by induction. One may check that ''f''_''n'' converges pointwise to the Cantor function defined above. Furthermore, the convergence is uniform. Indeed, separating into three cases, according to the definition of ''f''_''n''+1, 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 ''<math>f''</math> denotes the limit function, it follows that, for every ''<math>n''&nbsp;&ge;&nbsp; \geq 0</math>,

:<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>

=== 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,&nbsp;1] that do not contain the digit 1 in their [[radix|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 = \loglog_3(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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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 sameanalogous symmetry relations, althoughwith only ina anslightly altered form.

== Generalizations ==

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For ''z''&nbsp;=&nbsp;1/3, the inverse of the function ''x'' = 2&nbsp;''C''_1/3(''y'') is the Cantor function. That is, ''y''&nbsp;=&nbsp;''y''(''x'') is the Cantor function. In general, for any ''z''&nbsp;&lt;&nbsp;1/2, ''C''_''z''(''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 focuses 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/\log3log_3(2))^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| doi-broken-dateauthorlink=1Kenneth November 2024Falconer (mathematician)|bibcode = 2004MPCPS.136..167F|s2cid = 122381614}}</ref> showed that this squaring relationship holds for all Ahlfors'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.