Page "Cantor function" Paragraph 13

The Cantor function challenges naive intuitions about continuity and measure ; though it is continuous everywhere and has zero derivative almost everywhere, c goes from 0 to 1 as 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 ( and hence also continuous ) but not absolutely continuous.

It is constant on intervals of the form ( 0. x < sub > 1 </ sub > x < sub > 2 </ sub > x < sub > 3 </ sub >... x < sub > n </ sub > 022222 ..., 0. x < sub > 1 </ sub > x < sub > 2 </ sub > x < sub > 3 </ sub >... x < sub > n </ sub > 200000 ...), 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.