Discontinuous robust mappings are approximatable

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 cl A = cl int A . A mapping f: X Y is said tobe robust iffforeach open set Uy of Y, f-I(Uy) 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 points of continuity of f is dense in X and that for any other point x E X, (x, f(x)) is the limit of {(x , f(x,))}, where for all a, x, is a continuous point of f . This result justifies the notion of robustness.