Cantor function

In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. 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 (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 Harnack. The Cantor function was discussed and popularized by Scheeffer (1884), Lebesgue (1904) and Vitali (1905).

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:

For example:

The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, ${\textstyle c(x)}$ goes from 0 to 1 as ${\textstyle x}$ 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 = log 2/log 3) but not absolutely continuous. It is constant on intervals of the form (0.x₁x₂x₃...x_n022222..., 0.x₁x₂x₃...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.

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