Given a 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 connections between the following concepts in topological data analysis: the Euler characteristic transform (ECT), smooth ECT, Euler-Radon transform (ERT), and smooth ERT.

翻译：给定可定义集$S$上的可定义函数$f: S \to \mathbb{R}$，我们研究所有$t \in \mathbb{R}$时形如$S^f_t = \{x \in S: f(x) \leq t\}$的下水平集。利用o-极小结构，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，我们表明对于所有充分小的$\delta > 0$，$S^f_{t+\delta}$可形变收缩到$S^f_t$。应用这些结果，我们还刻画了拓扑数据分析中以下概念之间的联系：欧拉示性数变换（ECT）、光滑ECT、欧拉-拉东变换（ERT）与光滑ERT。