At which points exactly has Lebesgue's singular function the derivative zero ?

TL;DR

This paper characterizes the points where Lebesgue's singular function has a zero derivative, using binary expansion, and explores the differentiability of compositions involving Takagi's function.

Contribution

It provides a partial characterization of points with zero derivative for Lebesgue's singular function based on binary expansion analysis.

Findings

Identifies sets where the derivative of Lebesgue's singular function is zero.

Analyzes the differentiability of compositions with Takagi's function.

Offers insights into the structure of singular functions and their derivatives.

Abstract

Let L_a(x) be Lebesgue's singular function with a real parameter a (0<a<1, a not equal to 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 infinity? 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.