Cantor function: Difference between revisions

Revision as of 04:06, 26 March 2022

Revision as of 04:07, 26 March 2022

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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](https://en.wikipedia.org/wiki/[Pathological_(mathematics)#Pathological_example)|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>{{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}}.