The Thomae Function: Fractal Insights

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 θ > 0, it exhibits a rich self-similar
structure and intriguing regularity properties. After revisiting its fundamental characteristics, we analyze its Hölder 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.