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'[[File:CantorEscalier.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 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 after the first 1 by 0. #Replace all 2s with 1s. #Interpret the result as a binary number. The result is ''c''(''x''). For example: * 1/4 becomes 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 becomes 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 becomes 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. ==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''&nbsp;=&nbsp;log&nbsp;2/log&nbsp;3) 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 scheme|Bernoulli measure]] μ supported on the Cantor set: <math display="inline">c(x)=\mu([0,x])</math>. This 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. == Alternative definitions == === Iterative construction === [[File:Cantor function sequence.png|250px|right]] Below we define a sequence {''&fnof;''_n} of functions on the unit interval that converges to the Cantor function. Let ''&fnof;''₀(''x'') = ''x''. Then, for every integer {{nowrap|''n'' &ge; 0}}, the next function ''&fnof;''_''n''+1(''x'') will be defined in terms of ''&fnof;''_''n''(''x'') as follows: Let ''&fnof;''_''n''+1(''x'')&nbsp;= {{nowrap|0.5 &times; ''&fnof;''_''n''(3''x'')}},&nbsp; when {{nowrap|0 ≤ ''x'' ≤ 1/3&thinsp;}}; Let ''&fnof;''_''n''+1(''x'')&nbsp;= 0.5,&nbsp; when {{nowrap|1/3 ≤ ''x'' ≤ 2/3&thinsp;}}; Let ''&fnof;''_''n''+1(''x'')&nbsp;= {{nowrap|0.5 + 0.5 &times; ''&fnof;''_''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 ''&fnof;''_''n''(0)&nbsp;= 0 and ''&fnof;''_''n''(1)&nbsp;= 1 for every&nbsp;''n'', by induction. One may check that ''&fnof;''_''n'' converges pointwise to the Cantor function defined above. Furthermore, the convergence is uniform. Indeed, separating into three cases, according to the definition of ''&fnof;''_''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 ''&fnof;'' denotes the limit function, it follows that, for every ''n''&nbsp;&ge;&nbsp;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 ''&fnof;''₀(0)&nbsp;= 0, ''&fnof;''₀(1)&nbsp;= 1 and ''&fnof;''₀ 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,&nbsp;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> == 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''_''k'' &isin; {0,1}. Then consider the function : <math>C_z(y)=\sum_{k=1}^\infty b_k z^{k}.</math> 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 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 90s 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 its support, <math>(\log2/\log3)^2</math>. More recently [[Kenneth Falconer (mathematician)|Falconer]]<ref>{{Cite journal|title = One-sided multifractal analysis and points of non-differentiability of devil's staircases|url = http://journals.cambridge.org/article_S0305004103006960|journal = Mathematical Proceedings of the Cambridge Philosophical Society|date = 2004-01-01|issn = 1469-8064|pages = 167–174|volume = 136|issue = 01|doi = 10.1017/S0305004103006960|first = Kenneth J.|last = Falconer}}</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}\mathrm{supp}(\mu)\right)^2</math>Later, Troscheit<ref>{{Cite journal|title = Hölder differentiability of self-conformal devil's staircases|url = http://journals.cambridge.org/article_S0305004113000698|journal = Mathematical Proceedings of the Cambridge Philosophical Society|date = 2014-03-01|issn = 1469-8064|pages = 295–311|volume = 156|issue = 02|doi = 10.1017/S0305004113000698|first = Sascha|last = Troscheit}}</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. ==Notes== <references /> ==References== * {{cite book|ref=harv|first1=Richard Franklin|last1=Bass|author1-link=Richard F. Bass|title=Real analysis for graduate students|year=2013|origyear=2011|edition=Second|publisher=Createspace Independent Publishing|isbn=978-1-4818-6914-0}} *{{citation|first= G.|last= Cantor|title= De la puissance des ensembles parfaits de points|journal= Acta Math.|volume= 4 |year=1884|pages= 381–392|doi=10.1007/BF02418423}} Reprinted in: E. Zermelo (Ed.), Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts, Springer, New York, 1980. *{{citation|mr=2681574 |last=Darst|first= 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}} *{{citation |mr=2195181 |last=Dovgoshey |first=O. |last2=Martio |first2=O. |last3=Ryazanov |first3=V. |last4=Vuorinen |first4=M. |title=The Cantor function |journal=Expo. Math. |volume=24 |year=2006 |issue=1 |pages=1–37 |url=http://users.utu.fi/vuorinen/REA12/107.pdf }}{{dead link|date=November 2016 |bot=InternetArchiveBot |fix-attempted=yes }} *{{citation|first=J.F.|last= Fleron|title= A note on the history of the Cantor set and Cantor function|journal= Math. Mag.|volume= 67 |year=1994|pages= 136–140|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}} *Leoni, Giovanni (2017). ''[http://bookstore.ams.org/gsm-181/ A First Course in Sobolev Spaces: Second Edition]''. [[Graduate Studies in Mathematics]]. '''181'''. American Mathematical Society. pp. 734. '''{{ISBN|978-1-4704-2921-8}}''' *{{citation|first=Ludwig|last= Scheeffer|title= Allgemeine Untersuchungen über Rectification der Curven|journal= Acta Math.|volume= 5 |year=1884|pages= 49–82|doi=10.1007/BF02421552}} * {{cite book|ref=harv|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|origyear=2001|isbn=978-1-4348-4367-8}} *{{cite book|ref=harv|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}} == External links == * [http://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]]'