Editing Semi-continuity (section)

== Properties ==

Unless specified otherwise, all functions below are from a [[topological space]] <math>X</math> to the [[extended real number]]s <math>\overline{\R}= [-\infty,\infty].</math> ''Several of the results hold for semicontinuity at a specific point, but for brevity they are only stated for semicontinuity over the whole domain.''

* A function <math>f:X\to\overline{\R}</math> is [[Continuous function|continuous]] if and only if it is both upper and lower semicontinuous.
* The [[indicator function|characteristic function]] or [[indicator function]] of a set <math>A\subset X</math> (defined by <math>\mathbf{1}_A(x)=1</math> if <math>x\in A</math> and <math>0</math> if <math>x\notin A</math>) is upper semicontinuous if and only if <math>A</math> is a [[closed set]].  It is lower semicontinuous if and only if <math>A</math> is an [[open set]].
* In the field of [[convex analysis]], the [[Characteristic function (convex analysis)|characteristic function]] of a set <math>A \subset X</math> is defined differently, as <math>\chi_{A}(x)=0</math> if <math>x\in A</math> and <math>\chi_A(x) = \infty</math> if <math>x\notin A</math>. With that definition, the characteristic function of any {{em|closed set}} is lower semicontinuous, and the characteristic function of any {{em|open set}} is upper semicontinuous.

=== Binary operations on semicontinuous functions ===
Let <math>f,g : X \to \overline{\R}</math>.
* If <math>f</math> and <math>g</math> are lower semicontinuous, then the sum <math>f+g</math> is lower semicontinuous<ref>{{cite book|last1=Puterman|first1=Martin L.|title=Markov Decision Processes Discrete Stochastic Dynamic Programming|url=https://archive.org/details/markovdecisionpr00pute_298|url-access=limited|date=2005|publisher=Wiley-Interscience|isbn=978-0-471-72782-8|pages=[https://archive.org/details/markovdecisionpr00pute_298/page/n618 602]}}</ref> (provided the sum is well-defined, i.e., <math>f(x)+g(x)</math> is not the [[indeterminate form]] <math>-\infty+\infty</math>). The same holds for upper semicontinuous functions.
* If <math>f</math> and <math>g</math> are lower semicontinuous and non-negative, then the product function <math>f g</math> is lower semicontinuous.  The corresponding result holds for upper semicontinuous functions.
* The function <math>f</math> is lower semicontinuous if and only if <math>-f</math> is upper semicontinuous.
* If <math>f</math> and <math>g</math> are upper semicontinuous and <math>f</math> is  [[Monotonic function|non-decreasing]], then the [[Function composition|composition]] <math>f \circ g</math> is upper semicontinuous. On the other hand, if <math>f</math> is not non-decreasing, then <math>f \circ g</math> may not be upper semicontinuous. For example take <math>f : \R \to \R </math> defined as <math>f(x)=-x</math>. Then <math>f </math> is continuous and  <math>f \circ g = -g</math>, which is not upper semicontinuous unless <math>g</math> is continuous.
* If <math>f</math> and <math>g</math> are lower semicontinuous, their (pointwise) maximum and minimum (defined by <math>x \mapsto \max\{f(x), g(x)\}</math> and <math>x \mapsto \min\{f(x), g(x)\}</math>) are also lower semicontinuous.  Consequently, the set of all lower semicontinuous functions from <math>X</math> to <math>\overline{\R}</math> (or to <math>\R</math>) forms a [[lattice (order)|lattice]].  The corresponding statements also hold for upper semicontinuous functions.

=== Optimization of semicontinuous functions ===

* The (pointwise) [[supremum]] of an arbitrary family <math>(f_i)_{i\in I}</math> of lower semicontinuous functions <math>f_i:X\to\overline{\R}</math> (defined by <math>f(x)=\sup\{f_i(x):i\in I\}</math>) is lower semicontinuous.<ref>{{cite web |title=To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous |url=https://math.stackexchange.com/q/1662726}}</ref>

:In particular, the limit of a [[monotone increasing]] sequence <math>f_1\le f_2\le f_3\le\cdots</math> of continuous functions is lower semicontinuous.  (The Theorem of Baire below provides a partial converse.)  The limit function will only be lower semicontinuous in general, not continuous.  An example is given by the functions <math>f_n(x)=1-(1-x)^n</math> defined for <math>x\in[0,1]</math> for <math>n=1,2,\ldots.</math>

:Likewise, the [[infimum]] of an arbitrary family of upper semicontinuous functions is upper semicontinuous.  And the limit of a [[monotone decreasing]] sequence of continuous functions is upper semicontinuous.

* If <math>C</math> is a [[compact space]] (for instance a closed bounded interval <math>[a, b]</math>) and <math>f : C \to \overline{\R}</math> is upper semicontinuous, then <math>f</math> attains a maximum on <math>C.</math> If <math>f</math> is lower semicontinuous on <math>C,</math> it attains a minimum on <math>C.</math>
:(''Proof for the upper semicontinuous case'': By condition (5) in the definition, <math>f</math> is continuous when <math>\overline{\R}</math> is given the left order topology.  So its image <math>f(C)</math> is compact in that topology.  And the compact sets in that topology are exactly the sets with a maximum.  For an alternative proof, see the article on the [[extreme value theorem]].)

=== Other properties ===

* ('''Theorem of Baire''')<ref group="note">The result was proved by René Baire in 1904 for real-valued function defined on <math>\R</math>.  It was extended to metric spaces by [[Hans Hahn (mathematician)|Hans Hahn]] in 1917, and [[Hing Tong]] showed in 1952 that the most general class of spaces where the theorem holds is the class of [[perfectly normal space]]s.  (See Engelking, Exercise 1.7.15(c), p. 62 for details and specific references.)</ref> Let <math>X</math> be a [[metric space]].  Every lower semicontinuous function <math>f:X\to\overline{\R}</math> is the limit of a point-wise  [[monotone increasing|increasing]] sequence of extended real-valued continuous functions on <math>X.</math> In particular, there exists a sequence <math>\{f_i\}</math> of continuous functions <math>f_i : X \to \overline\R</math> such that 
:<math display="block">f_i(x) \leq f_{i+1}(x) \quad \forall x \in X,\ \forall i = 0, 1, 2, \dots</math> and 
:<math display="block">\lim_{i \to \infty} f_i(x) = f(x) \quad \forall x \in X. </math>
:If <math>f</math> does not take the value <math>-\infty</math>, the continuous functions can be taken to be real-valued.<ref>Stromberg, p. 132, Exercise 4(g)</ref><ref>{{cite web |title=Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions |url=https://math.stackexchange.com/q/1279763}}</ref>

:Additionally, every upper semicontinuous function <math>f:X\to\overline{\R}</math> is the limit of a [[monotone decreasing]] sequence of extended real-valued continuous functions on <math>X;</math> if <math>f</math> does not take the value <math>\infty,</math> the continuous functions can be taken to be real-valued.

* Any upper semicontinuous function <math>f : X \to \N</math> on an arbitrary topological space <math>X</math> is locally constant on some [[dense set|dense open subset]] of <math>X.</math>

* If the topological space <math>X</math> is [[Sequential_space|sequential]], then <math>f : X \to \mathbb{R}</math> is upper semi-continuous if and only if it is sequentially upper semi-continuous, that is, if for any <math>x \in X</math> and any sequence <math>(x_n)_n \subset X</math> that converges towards <math>x</math>, there holds <math>\limsup_{n \to \infty} f(x_n) \leqslant f(x)</math>. Equivalently, in a sequential space, <math>f</math> is upper semicontinuous if and only if its superlevel sets <math>\{\, x \in X \,|\, f(x) \geqslant y \,\}</math> are [[Fréchet–Urysohn_space#Sequentially_open%2Fclosed_sets|sequentially closed]] for all <math>y \in \mathbb{R}</math>. In general, upper semicontinuous functions are sequentially upper semicontinuous, but the converse may be false.