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Lower Semicontinuous Functions
| Title: | Lower Semicontinuous Functions |
| Author: | Bogdan Grechuk (grechukbogdan /at/ yandex /dot/ ru) |
| Submission date: | 2011-01-08 |
| Abstract: | We define the notions of lower and upper semicontinuity for functions from a metric space to the extended real line. We prove that a function is both lower and upper semicontinuous if and only if it is continuous. We also give several equivalent characterizations of lower semicontinuity. In particular, we prove that a function is lower semicontinuous if and only if its epigraph is a closed set. Also, we introduce the notion of the lower semicontinuous hull of an arbitrary function and prove its basic properties. |
| BibTeX: |
@article{Lower_Semicontinuous-AFP,
author = {Bogdan Grechuk},
title = {Lower Semicontinuous Functions},
journal = {Archive of Formal Proofs},
month = jan,
year = 2011,
note = {\url{http://isa-afp.org/entries/Lower_Semicontinuous.shtml},
Formal proof development},
ISSN = {2150-914x},
}
|
| License: | BSD License |
| Status: [skipped] | This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations. |
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