Stability of closedness of semi-algebraic sets under continuous semi-algebraic mappings

Given a closed semi-algebraic set $X\subset {\mathbb{R}}^n$ and a continuous semi-algebraic mapping $G\text{\hspace{0.17em}⁣}:X\to {\mathbb{R}}^m,$ it will be shown that there exists an open dense semi-algebraic subset $\mathcal{U}$ of $L({\mathbb{R}}^n,{\mathbb{R}}^m),$ the space of all linear mappings from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m,$ such that for all $F\in \mathcal{U},$ the image $(F+G)(X)$ is a closed (semi-algebraic) set in ${\mathbb{R}}^m\mathrm{.}$ To do this, we study the tangent cone at infinity $C_{\mathrm{\infty }}X$ and the set $E_\infty X \subset C_\infty X$of(unit)exceptionaldirectionsatinfinityof$X.$Specificallyweshowthattheset$E_\infty X$isnowheredensein$C_\infty X \cap $\mathbb{S}$^{n - 1}.$