Tubes about Functions and Multifunctions

Abstract

We provide a characterization of lower semicontinuity for multifunctions with values in a metric space \(\langle Y,d \rangle\) which, in the special case of single-valued functions, says that a function is continuous if and only if for each \(\varepsilon \gt 0\), the \(\varepsilon\)-tube about its graph is an open set. Applications are given, one of which provides a novel understanding of the Open Mapping Theorem from functional analysis. We also give a related but more complicated characterization of upper semicontinuity for multifunctions with closed values in a metrizable space.