## Real Analysis Exchange

### 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.

#### Article information

Source
Real Anal. Exchange, Volume 39, Number 1 (2013), 33-44.

Dates
First available in Project Euclid: 1 July 2014