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In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious 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.

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rdf:type	building
rdfs:label	Cantor function (en)
rdfs:comment	In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious 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. (en)
sameAs	Cantor function Cantor function
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Subject	De Rham curves Special functions Fractals Georg Cantor Measure theory
Link from a Wikipage to an external page	http://bookstore.ams.org/gsm-181/ http://demonstrations.wolfram.com/CantorFunction/ https://www.encyclopediaofmath.org/index.php/Cantor_ternary_function
thumbnail	wiki-commons:Special:FilePath/CantorEscalier.svg?width=300
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gold:hypernym	Example
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Wikipage revision ID	1071592515 (xsd:integer)
Link from a Wikipage to another Wikipage	Fractal Fractal dimension Probability density function Uncountable set Uniform continuity Absolute continuity Almost everywhere Arc length Atom (measure theory) Base (exponentiation) Bernoulli process Binary tree Monoid Self-similarity Bounded function Cantor distribution Cantor set Carl Gustav Axel Harnack De Rham curves Special functions Fractals Georg Cantor Measure theory Disjoint sets Wolfram Demonstrations Project Cumulative distribution function De Rham curve Hölder condition Concatenation Continuous function Counterexample Dyadic rational Dyadic transformation Hermann Minkowski Minkowski's question-mark function Function (mathematics) Fundamental theorem of calculus Kenneth Falconer (mathematician) Symmetry Derivative Hausdorff dimension Modular group Lebesgue measure Mathematics Measure (mathematics) Singular function
has abstract	In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious 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, Lebesgue's singular function, the Cantor–Vitali function, the Devil's staircase, the Cantor staircase function, and the Cantor–Lebesgue function. Georg Cantor 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 Harnack. The Cantor function was discussed and popularized by , and . (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Cantor_function
is Wikipage redirect of	Cantor-Lebesgue function Cantor staircase Cantor ternary function Cantor–Lebesgue function Lebesgue singular function
is Link from a Wikipage to another Wikipage of	Fractal string Null set 1884 in science Absolute continuity Almost all Arnold tongue Bernoulli process Blancmange curve Monotonic function Cantor (disambiguation) Cantor distribution Cantor set Cantor space

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