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2013/2014 Tubes about Functions and Multifunctions
Gerald Beer, Michael J. Hoffman
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Real Anal. Exchange 39(1): 33-44 (2013/2014).


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.


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Gerald Beer. Michael J. Hoffman. "Tubes about Functions and Multifunctions." Real Anal. Exchange 39 (1) 33 - 44, 2013/2014.


Published: 2013/2014
First available in Project Euclid: 1 July 2014

zbMATH: 1301.54034
MathSciNet: MR3261897

Primary: 54C05 , ‎54C60‎
Secondary: 46A30 , 54E35

Keywords: Continuous function , open mapping , Open Mapping Theorem , semicontinuous function , semicontinuous multifunction , tube , Uniform convergence

Rights: Copyright © 2013 Michigan State University Press

Vol.39 • No. 1 • 2013/2014
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