A lower semicontinuous regularization for set-valued mappings and its applications

A basic fact in real analysis is that every real-valued function f admits a lower semicontinuous regularization (f) under bar, defined by means of the lower Emit of f: (f) under bar (x) := lim inf f(y -> x) (y). This fact breaks down for set-valued mappings. In this note, we first provide some counterexamples. We try further to define a kind of lower semicontinuous regularization for a given set-valued mapping and we point out some general applications.