Given a continuous definable function $f: S \to \mathbb{R}$ on a definable set $S$, we study sublevel sets of the form $S^f_t = \{x \in S: f(x) \leq t\}$ for all $t \in \mathbb{R}$. Using o-minimal structures, we prove that the Euler characteristic of $S^f_t$ is right continuous with respect to $t$. Furthermore, when $S$ is compact, we show that $S^f_{t+\delta}$ deformation retracts to $S^f_t$ for all sufficiently small $\delta > 0$. Applying these results, we also characterize the relationship between the concepts of Euler characteristic transform and smooth Euler characteristic transform in topological data analysis.

翻译：给定可定义集$S$上的连续可定义函数$f: S \to \mathbb{R}$，我们研究形如$S^f_t = \{x \in S: f(x) \leq t\}$（$t \in \mathbb{R}$）的子水平集。利用o-minimal结构，我们证明了$S^f_t$的欧拉示性数关于$t$是右连续的。进一步地，当$S$为紧集时，我们展示了对于所有足够小的$\delta > 0$，$S^f_{t+\delta}$可形变收缩至$S^f_t$。应用这些结果，我们还刻画了拓扑数据分析中欧拉示性数变换与光滑欧拉示性数变换概念之间的关系。