Cantor function

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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. Although it is continuous everywhere, and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument goes from 0 to 1. Thus, while the function seems like a constant one that cannot grow, it does indeed monotonically grow. It is also called 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).