On the set of points where Lebesgue’s singular function has the derivative zero

Abstract

Let $L_{a}(x)$ be Lebesgue’s singular function with a real parameter $a$ ($0<a<1, a \neq 1/2$). As is well known, $L_{a}(x)$ is strictly increasing and has a derivative equal to zero almost everywhere. However, what sets of $x \in [0,1]$ actually have $L_{a}'(x)=0$ or $+\infty$? We give a partial characterization of these sets in terms of the binary expansion of $x$. As an application, we consider the differentiability of the composition of Takagi’s nowhere differentiable function and the inverse of Lebesgue’s singular function.