Real Analysis Exchange

Tubes about Functions and Multifunctions

Gerald Beer and Michael J. Hoffman

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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

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

First available in Project Euclid: 1 July 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54C05: Continuous maps 54C60: Set-valued maps [See also 26E25, 28B20, 47H04, 58C06]
Secondary: 54E35: Metric spaces, metrizability 46A30: Open mapping and closed graph theorems; completeness (including $B$-, $B_r$-completeness)

tube semicontinuous multifunction semicontinuous function continuous function uniform convergence open mapping Open Mapping Theorem


Beer, Gerald; Hoffman, Michael J. Tubes about Functions and Multifunctions. Real Anal. Exchange 39 (2013), no. 1, 33--44.

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