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Given a closed semi-algebraic set $X \subset \mathbb{R}^n$ and a continuoussemi-algebraic mapping $G \colon X \to \mathbb{R}^m,$ it will be shown thatthere exists an open dense semi-algebraic subset $\mathscr{U}$ 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 $F \in \mathscr{U},$ theimage $(F + G)(X)$ is a closed (semi-algebraic) set in $\mathbb{R}^m.$ To dothis, we study the tangent cone at infinity $C_\infty X$ and the set $E_\inftyX \subset C_\infty X$ of (unit) exceptional directions at infinity of $X.$Specifically we show that the set $E_\infty X$ is nowhere dense in $C_\infty X\cap \mathbb{S}^{n - 1}.$

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