DISCONTINUOUS ROBUST MAPPINGS ARE APPROXIMATABLE

The concepts of robustness of sets and and functions were introduced t o form the foundation of the theory of integral global optimization. A set A of a topological space X is said to be robust iff cl A = cl int A. A mapping f: X --> 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 --> Y is robust iff it is approximatable in the sense that the set of poin ts of continuity of f is dense in X and that for any other point x is an element of 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.