Discontinuous robust mappings are approximatable

Abstract:

The concepts of robustness of sets and and functions were introduced to form the foundation of the theory of integral global optimization. A set $A$ of a topological space $X$ is said to be robust iff ${\text {cl}}A = {\text {cl}}$ int $A$. A mapping $f:X \to Y$ is said to be robust iff for each open set ${U_Y}$ of $Y$, ${f^{ - 1}}({U_Y})$ is robust. We prove that if $X$ is a Baire space and $Y$ satisfies the second axiom of countability, then a mapping $f:X \to Y$ is robust iff it is approximatable in the sense that the set of points of continuity of $f$ is dense in $X$ and that for any other point $x \in X$, $(x,f(x))$ is the limit of $\{ ({x_\alpha },f({x_\alpha }))\}$, where for all $\alpha$, ${x_\alpha }$ is a continuous point of $f$. This result justifies the notion of robustness.