Functions with Continuous Upper and Lower Envelopes

Functions with Continuous Upper and Lower Envelopes

Given a real-valued upper semicontinuous function h and a real-valued lower semicontinuous function g on a metric space such that (1) h ≥ g pointwise and (2) h ( x ) = g ( x ) at each isolated point of the space, it is not in general possible to find a real-valued function f whose upper envelope is...

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Veröffentlicht in:Vietnam journal of mathematics 2018-03, Vol.46 (1), p.169-175
1. Verfasser: Beer, Gerald
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Sprache:eng
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Continuity (mathematics)
Mathematical functions
Mathematics
Mathematics and Statistics
Metric space
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container_title Vietnam journal of mathematics
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creator Beer, Gerald
description Given a real-valued upper semicontinuous function h and a real-valued lower semicontinuous function g on a metric space such that (1) h ≥ g pointwise and (2) h ( x ) = g ( x ) at each isolated point of the space, it is not in general possible to find a real-valued function f whose upper envelope is h and whose lower envelope is g , even if the space is compact and dense-in-itself. The purpose of this note is to show that such an f exists in the case that both h and g are continuous, and that f can be chosen to be a Borel function.
doi_str_mv 10.1007/s10013-017-0266-7
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subjects Continuity (mathematics)
Mathematical functions
Mathematics
Mathematics and Statistics
Metric space
title Functions with Continuous Upper and Lower Envelopes
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