# Infinite derivatives of the Takagi-Van der Waerden functions

## TL;DR

This paper investigates the Takagi-Van der Waerden functions, characterizing points with infinite lateral derivatives and establishing that this set has Hausdorff dimension one but zero Lebesgue measure.

## Contribution

It provides a detailed characterization of the points with infinite derivatives and proves the measure-theoretic properties of this set.

## Key findings

- Set of points with infinite derivatives has Hausdorff dimension one.
- This set has Lebesgue measure zero.
- Lateral derivatives are infinite at these points.

## Abstract

In this paper we characterize the set of points where the lateral derivatives of the Takagi-Van der Waerden functions are infinite. We also prove that the set of points with infinite derivative has Hausdorff dimension one and Lebesgue measure zero.