Cantor function: Difference between revisions

Revision as of 21:13, 21 September 2019

Revision as of 21:15, 21 September 2019

Line 2:

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

[[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 known as a notorious counter-example in analysis, because it challenges naive intuitions about continuity and measure. Though it is continuous everywhere and has zero derivative almost everywhere, {\textstyle c(x)}it goes from 0 to 1 as {\textstylethe x}argument goes from 0 to 1. The function even takes on every value in between. Thus, in one sense itthe function seems very much like a constant functionone which cannot grow, and in another, it does indeed monotonously 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}}.