  We prove a version of the Thom Isotopy Theorem for nonproper semialgebraic
maps $f\colon X\rightarrow \mathbb{R}^m$, where $X \subset\mathbb{R}^n$ is a
semialgebraic set and $f$ is the restriction to $X$ of a smooth semialgebraic
map $F:\mathbb{R}^n\to \mathbb{R}^m$.
