Editing Semi-continuity (section)

{{short description|Property of functions which is weaker than continuity}}
{{For|the notion of upper or lower semi-continuous [[set-valued function]]|Hemicontinuity}}

In [[mathematical analysis]], '''semicontinuity''' (or '''semi-continuity''') is a property of [[Extended real number|extended real]]-valued [[Function (mathematics)|function]]s that is weaker than [[Continuous function|continuity]]. An extended real-valued function <math>f</math> is '''upper''' (respectively, '''lower''') '''semicontinuous''' at a point <math>x_0</math> if, roughly speaking, the function values for arguments near <math>x_0</math> are not much higher (respectively, lower) than <math>f\left(x_0\right).</math> Briefly, a function on a domain <math>X</math> is lower semi-continuous if its [[epigraph (mathematics)|epigraph]] <math>\{(x,t)\in X\times\R : t\ge f(x)\}</math> is closed in <math>X\times\R</math>, and upper semi-continuous if <math>-f</math> is lower semi-continuous.

A function is continuous if and only if it is both upper  and lower semicontinuous. If we take a continuous function and increase its value at a certain point <math>x_0</math> to <math>f\left(x_0\right) + c</math> for some <math>c>0</math>, then the result is upper semicontinuous; if we decrease its value to <math>f\left(x_0\right) - c</math> then the result is lower semicontinuous.

[[File:Upper semi.svg|thumb|right|An upper semicontinuous function that is not lower semicontinuous at <math>x_0</math>. The solid blue dot indicates <math>f\left(x_0\right).</math>]] [[File:Lower semi.svg|thumb|right|A lower semicontinuous function that is not upper semicontinuous at <math>x_0</math>.  The solid blue dot indicates <math>f\left(x_0\right).</math>]]

The notion of upper and lower semicontinuous function was first introduced and studied by [[René Baire]] in his thesis in 1899.<ref>{{cite web |last1=Verry |first1=Matthieu |title=Histoire des mathématiques - René Baire |url=https://www.researchgate.net/publication/351274714}}</ref>