Points of infinite derivative of Cantor functions

Abstract

We consider self-similar Borel probability measures $\mu $ on a self-similar set $E$ with strong separation property. We prove that for any such measure $\mu $ the derivative of its distribution function $F(x)$ is infinite for $\mu $-a.e. $x\in E$, and so the set of points at which $F(x)$ has no derivative, finite or infinite is of $\mu $-zero.