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.
  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.