Abstract :
[en] The Thomae function has long served as a striking example in real analysis, showcasing the interplay between continuity and discontinuity. Introduced by Thomae in 1875 as a refinement of the Dirichlet function, within the framework of Riemann's concept of integration, it is defined as follows. Unless explicitly stated otherwise, any rational number x expressed as x=p/q (p in Z, q in N) with p and q coprime. The Thomae function is then given by T_θ (x) = 1 if x=0, T_θ (x) = q^(-θ) if x is rational with x=p/q and T_θ (x) = 0 if x is irrational, with θ=1. The limiting case θ=0 corresponds to the Dirichlet function. For θ>0, the function exhibits the remarkable property of being continuous on the irrational numbers while discontinuous at every rational point. This duality, combined with its self-similar structure, renders the Thomae function an essential object of study for understanding irregular functions in analysis. Here, we will focus on the case where θ in (0,2]. Beyond its classical role in real analysis, the Thomae function has found relevance in broader mathematical and applied contexts. Recent studies have highlighted analogies between its spiked structure and distributions observed in empirical datasets, particularly in biology and clinical research.
This talk focuses on the Hölder regularity of the Thomae function, a key aspect of its behavior. First, we review its fundamental properties, offering a detailed account of its defining characteristics and self-similar nature. Then, we analyze the function's Hölder regularity, uncovering insights into its fractal-like properties through contemporary mathematical tools. By bridging its classical foundations with these contemporary perspectives, we aim to highlight both the theoretical elegance and the deeper structural nuances of this remarkable function.
