Cantor function: Difference between revisions

Revision as of 01:16, 17 January 2019

Revision as of 01:17, 17 January 2019

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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-VitaliCantor–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-LebesgueCantor–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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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 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.