A Lower Semicontinuous Regularization for Set-Valued Mappings and its Applications

A basic fact in real analysis is that every real-valued function

admits a lower semicontinuous regularization

\underline{f} f ‾

, defined by means of the lower limit of

f f

:

\begin{aligned}\underline{f}\left( x\right):=\;\displaystyle\liminf_{y\rightarrow x}f\left( y\right).\end{aligned} f ‾ ( x ) : =    lim inf ⁡ y → x f ( 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.