The Thomae Function: Fractal Insights

TL;DR

This paper explores the fractal and regularity properties of the Thomae function, highlighting its self-similar structure and analyzing its continuity and irregularity using classical analysis tools.

Contribution

It offers a detailed analysis of the Thomae function's fractal structure and continuity properties, providing new insights into its irregularity and regularity on dense subsets.

Findings

Thomae function exhibits rich self-similarity and fractal structure.

It is continuous on irrationals and discontinuous on rationals.

The function's H"older continuity varies with parameters.

Abstract

This article examines the Thomae function, a paradigmatic example of a function that is continuous on the irrationals and discontinuous elsewhere. Defined for a parameter $\theta>0$, it exhibits a rich self-similar structure and intriguing regularity properties. After revisiting its fundamental characteristics, we analyze its H\"older continuity, emphasizing the interplay between its discrete spikes and its behavior on dense subsets of the real line. This study provides a refined perspective on the irregularity of the Thomae function, using classical analytical tools to elucidate its fractal nature.