A characterization of the nowhere differentiable functions of the Generalized Takagi Class

TL;DR

This paper characterizes when functions in certain Generalized Takagi Classes, including the Takagi-Van der Waerden Class, are nowhere differentiable, linking this property to the behavior of their weight sequences.

Contribution

It provides a precise criterion for nowhere differentiability in these classes based on the sequence of weights, extending understanding of fractal and pathological functions.

Findings

Functions are nowhere differentiable iff weights do not tend to zero.

Characterization applies specifically to the Takagi-Van der Waerden Class.

Links weight sequence behavior to differentiability properties.

Abstract

In this paper, we prove that for some Generalized Takagi Classes, in particular for the Takagi-Van der Waerden Class, the functions are nowhere differentiable if, and only if, the sequence of weights does not belong to $c_0$.