Editing Semi-continuity (section)

== Definitions ==

Assume throughout that <math>X</math> is a [[topological space]] and <math>f:X\to\overline{\R}</math> is a function with values in the [[extended real number]]s <math>\overline{\R}=\R \cup \{-\infty,\infty\} = [-\infty,\infty]</math>.

=== Upper semicontinuity ===

A function <math>f:X\to\overline{\R}</math> is called '''upper semicontinuous at a point''' <math>x_0 \in X</math> if for every real <math>y > f\left(x_0\right)</math> there exists a [[neighborhood (topology)|neighborhood]] <math>U</math> of <math>x_0</math> such that <math>f(x)<y</math> for all <math>x\in U</math>.<ref name="Stromberg">Stromberg, p. 132, Exercise 4</ref>
Equivalently, <math>f</math> is upper semicontinuous at <math>x_0</math> if and only if
<math display=block>\limsup_{x \to x_0} f(x) \leq f(x_0)</math>
where lim sup is the [[limit superior (topological space)|limit superior]] of the function <math>f</math> at the point <math>x_0.</math>

If <math>X</math> is a [[metric space]] with [[distance function]] <math>d</math> and <math>f(x_0)\in\R,</math> this can also be restated using an <math>\varepsilon</math>-<math>\delta</math> formulation, similar to the definition of [[continuous function]].  Namely, for each <math>\varepsilon>0</math> there is a <math>\delta>0</math> such that <math>f(x)<f(x_0)+\varepsilon</math> whenever <math>d(x,x_0)<\delta.</math>

A function <math>f:X\to\overline{\R}</math> is called '''upper semicontinuous''' if it satisfies any of the following equivalent conditions:<ref name="Stromberg" />
:(1) The function is upper semicontinuous at every point of its [[domain (function)|domain]].
:(2) For each <math>y\in\R</math>, the set <math>f^{-1}([ -\infty ,y))=\{x\in X : f(x)<y\}</math> is [[open (topology)|open]] in <math>X</math>, where <math>[ -\infty ,y)=\{t\in\overline{\R}:t<y\}</math>.
:(3) For each <math>y\in\R</math>, the <math>y</math>-[[superlevel set]] <math>f^{-1}([y, \infty)) = \{x\in X : f(x)\ge y\}</math> is [[closed (topology)|closed]] in <math>X</math>.
:(4) The [[hypograph (mathematics)|hypograph]] <math>\{(x,t)\in X\times\R : t\le f(x)\}</math> is closed in <math>X\times\R</math>.
:(5) The function <math>f</math> is continuous when the [[codomain]] <math>\overline{\R}</math> is given the [[left order topology]].  This is just a restatement of condition (2) since the left order topology is generated by all the intervals <math>[ -\infty,y)</math>.

=== Lower semicontinuity ===

A function <math>f:X\to\overline{\R}</math> is called '''lower semicontinuous at a point''' <math>x_0\in X</math> if for every real <math>y < f\left(x_0\right)</math> there exists a [[neighborhood (topology)|neighborhood]] <math>U</math> of <math>x_0</math> such that <math>f(x)>y</math> for all <math>x\in U</math>.
Equivalently, <math>f</math> is lower semicontinuous at <math>x_0</math> if and only if
<math display=block>\liminf_{x \to x_0} f(x) \ge f(x_0)</math>
where <math>\liminf</math> is the [[limit inferior (topological space)|limit inferior]] of the function <math>f</math> at point <math>x_0.</math>

If <math>X</math> is a [[metric space]] with [[distance function]] <math>d</math> and <math>f(x_0)\in\R,</math> this can also be restated as follows: For each <math>\varepsilon>0</math> there is a <math>\delta>0</math> such that <math>f(x)>f(x_0)-\varepsilon</math> whenever <math>d(x,x_0)<\delta.</math>

A function <math>f:X\to\overline{\R}</math> is called '''lower semicontinuous''' if it satisfies any of the following equivalent conditions:
:(1) The function is lower semicontinuous at every point of its [[domain (function)|domain]].
:(2) For each <math>y\in\R</math>, the set <math>f^{-1}((y,\infty ])=\{x\in X : f(x)>y\}</math> is [[open (topology)|open]] in <math>X</math>, where <math>(y,\infty ]=\{t\in\overline{\R}:t>y\}</math>.
:(3) For each <math>y\in\R</math>, the <math>y</math>-[[sublevel set]] <math>f^{-1}((-\infty, y]) = \{x\in X : f(x)\le y\}</math> is [[closed (topology)|closed]] in <math>X</math>.
:(4) The [[epigraph (mathematics)|epigraph]] <math>\{(x,t)\in X\times\R : t\ge f(x)\}</math> is closed in <math>X\times\R</math>.<ref name="Kurdila2005">{{cite book | vauthors=((Kurdila, A. J.)), ((Zabarankin, M.)) | date= 2005 | chapter=Convex Functional Analysis | title=Lower Semicontinuous Functionals | publisher=Birkhäuser-Verlag | series=Systems & Control: Foundations & Applications | edition=1st | pages=205–219 | url=http://link.springer.com/10.1007/3-7643-7357-1_7 | doi=10.1007/3-7643-7357-1_7 | isbn=978-3-7643-2198-7}}</ref>{{rp|207}}
:(5) The function <math>f</math> is continuous when the [[codomain]] <math>\overline{\R}</math> is given the [[right order topology]].  This is just a restatement of condition (2) since the right order topology is generated by all the intervals <math>(y,\infty ] </math>.