Title: The Thomae Function: Fractal Insights Show Chinese title

Title: 托马斯函数：分形见解

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. Show Chinese abstract

Abstract: 本文研究了Thomae函数，这是一个典型的函数，在无理数上连续而在其他地方不连续。 对于参数$\theta>0$定义，它表现出丰富的自相似结构和引人入胜的规律性特征。 在回顾其基本特性之后，我们分析了它的Hölder连续性，强调其离散尖峰与其在实数线稠密子集上的行为之间的相互作用。 这项研究使用经典的分析工具，对Thomae函数的不规则性提供了更精细的视角，阐明了其分形性质。