# 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{https://isa-afp.org/entries/Lower_Semicontinuous.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Status: [ok] This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.