Differentiability of functions in the Zygmund class

TL;DR

This paper investigates the Hausdorff dimension of points where Zygmund class functions have bounded divided differences, establishing lower bounds and demonstrating sharpness of these bounds.

Contribution

It proves that the Hausdorff dimension of such points is at least 1 for Zygmund class functions and extends the result to Small Zygmund class functions with a constructed example.

Findings

Hausdorff dimension of points with bounded divided differences ≥ 1

Results are sharp, as shown by a constructed example

Extends to Small Zygmund class functions

Abstract

We prove that the Hausdorff dimension of the set of points where a function in the Zygmund class in the euclidean space has bounded divided differences, is bigger or equal to 1. A similar result for functions in the Small Zygmund class is also proved and an example is constructed to show that these results are sharp.