Editing Semi-continuity

{{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>

== 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>.

== Examples ==
Consider the function <math>f,</math> [[piecewise]] defined by:
<math display=block>f(x) = \begin{cases}
-1   & \mbox{if } x < 0,\\
 1   & \mbox{if } x \geq 0
\end{cases}</math>
This function is upper semicontinuous at <math>x_0 = 0,</math> but not lower semicontinuous.

The [[floor function]] <math>f(x) = \lfloor x \rfloor,</math> which returns the greatest integer less than or equal to a given real number <math>x,</math> is everywhere upper semicontinuous. Similarly, the [[ceiling function]] <math>f(x) = \lceil x \rceil</math> is lower semicontinuous.

Upper and lower semicontinuity bear no relation to [[Continuous function#Directional and semi-continuity|continuity from the left or from the right]] for functions of a real variable. Semicontinuity is defined in terms of an ordering in the range of the functions, not in the domain.<ref>Willard, p. 49, problem 7K</ref>  For example the function
<math display=block>f(x) = \begin{cases}
\sin(1/x) & \mbox{if } x \neq 0,\\
1         & \mbox{if } x = 0,
\end{cases}</math>
is upper semicontinuous at <math>x = 0</math> while the function limits from the left or right at zero do not even exist.

If <math>X = \R^n</math> is a Euclidean space (or more generally, a metric space) and <math>\Gamma = C([0,1], X)</math> is the space of [[curve]]s in <math>X</math> (with the [[Supremum norm|supremum distance]] <math>d_\Gamma(\alpha,\beta) = \sup\{d_X(\alpha(t),\beta(t)):t\in[0,1]\}</math>), then the length functional <math>L : \Gamma \to [0, +\infty],</math> which assigns to each curve <math>\alpha</math> its [[Curve#Length of curves|length]] <math>L(\alpha),</math> is lower semicontinuous.<ref>{{Cite book |last=Giaquinta |first=Mariano |url=https://www.worldcat.org/oclc/213079540 |title=Mathematical analysis : linear and metric structures and continuity |date=2007 |publisher=Birkhäuser |others=Giuseppe Modica |isbn=978-0-8176-4514-4 |edition=1 |location=Boston |at=Theorem 11.3, p.396 |oclc=213079540}}</ref> As an example, consider approximating the unit square diagonal by a staircase from below. The staircase always has length 2, while the diagonal line has only length <math>\sqrt 2</math>.

== 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.

== Semicontinuity of set-valued functions ==
For [[set-valued function]]s, several concepts of semicontinuity have been defined, namely ''upper'', ''lower'', ''outer'', and ''inner'' semicontinuity, as well as ''upper'' and ''lower [[hemicontinuity]]''.
A set-valued function <math>F</math> from a set <math>A</math> to a set <math>B</math> is written <math>F : A \rightrightarrows B.</math> For each <math>x \in A,</math> the function <math>F</math> defines a set <math>F(x) \subset B.</math>
The [[preimage]] of a set <math>S \subset B</math> under <math>F</math> is defined as 
<math display="block">F^{-1}(S) :=\{x \in A: F(x) \cap S \neq \varnothing\}.</math>
That is, <math>F^{-1}(S)</math> is the set that contains every point <math>x</math> in <math>A</math> such that <math>F(x)</math> is not [[Disjoint sets|disjoint]] from <math>S</math>.<ref name="freemanRobust"/>

=== Upper and lower semicontinuity ===

A set-valued map <math>F: \mathbb{R}^m \rightrightarrows \mathbb{R}^n</math> is ''upper semicontinuous'' at <math>x \in \mathbb{R}^m</math> if for every open set <math>U \subset \mathbb{R}^n</math> such that <math>F(x) \subset U</math>, there exists a neighborhood <math>V</math> of <math>x</math> such that <math>F(V) \subset U.</math><ref name="freemanRobust"/>{{rp|Def. 2.1}}

A set-valued map <math>F: \mathbb{R}^m \rightrightarrows \mathbb{R}^n</math> is ''lower semicontinuous'' at <math>x \in \mathbb{R}^m</math> if for every open set <math>U \subset \mathbb{R}^n</math> such that <math>x \in F^{-1}(U),</math> there exists a neighborhood <math>V</math> of <math>x</math> such that <math>V \subset F^{-1}(U).</math><ref name="freemanRobust"/>{{rp|Def. 2.2}}

Upper and lower set-valued semicontinuity are also defined more generally for a set-valued maps between topological spaces by replacing <math>\mathbb{R}^m</math> and <math>\mathbb{R}^n</math> in the above definitions with arbitrary topological spaces.<ref name="freemanRobust"/>

Note, that there is not a direct correspondence between single-valued lower and upper semicontinuity and set-valued lower and upper semicontinuouty. 
An upper semicontinuous single-valued function is not necessarily upper semicontinuous when considered as a set-valued map.<ref name="freemanRobust"/>{{rp|18}} 
For example, the function <math>f : \mathbb{R} \to \mathbb{R}</math> defined  by 
<math display=block>f(x) = \begin{cases}
-1   & \mbox{if } x < 0,\\
 1   & \mbox{if } x \geq 0
\end{cases}</math>
is upper semicontinuous in the single-valued sense but the set-valued map <math>x \mapsto F(x) := \{f(x)\}</math> is not upper semicontinuous in the set-valued sense. 
<!--
Conversely, an upper semicontinuous set-valued function (in the set-valued sense) may not be upper semicontinuous in the single-valued sense sense it may not be single-valued.
-->

=== Inner and outer semicontinuity ===

A set-valued function <math>F: \mathbb{R}^m \rightrightarrows \mathbb{R}^n</math> is called ''inner semicontinuous'' at <math>x</math> if for every <math>y \in F(x)</math> and every convergent sequence <math>(x_i)</math> in <math>\mathbb{R}^m</math> such that <math>x_i \to x</math>, there exists  
a sequence <math>(y_i)</math> in <math>\mathbb{R}^n</math> such that <math>y_i \to y</math> and <math>y_i \in F\left(x_i\right)</math> for all sufficiently large <math>i \in \mathbb{N}.</math><ref name="goebelSetvalued"/><ref group="note">In particular, there exists <math>i_0 \geq 0</math> such that <math>y_i \in F(x_i)</math> for every natural number <math>i \geq i_0,</math>. The  necessisty of only considering the tail of <math>y_i</math> comes from the fact that for small values of <math>i,</math> the set <math>F(x_i)</math> may be empty.</ref>

A set-valued function <math>F: \mathbb{R}^m \rightrightarrows \mathbb{R}^n</math> is called ''outer semicontinuous'' at <math>x</math> if for every convergence sequence <math>(x_i)</math> in <math>\mathbb{R}^m</math> such that <math>x_i \to x</math> and every convergent sequence <math>(y_i)</math> in <math>\mathbb{R}^n</math> such that <math>y_i \in F(x_i)</math> for each <math>i\in\mathbb{N},</math> the sequence <math>(y_i)</math> converges to a point in <math>F(x)</math> (that is, <math>\lim _{i \to \infty} y_i \in F(x)</math>).<ref name="goebelSetvalued"/>
<!--The definitions of upper and lower semicontinuity are defined using open neighborhoods, where as inner and outer semicontinuity are defined using convergent sequences.-->

== See also ==

* {{annotated link|left-continuous|Directional continuity}}
* {{annotated link|Katětov–Tong insertion theorem}}
* {{annotated link|Hemicontinuity}}
* {{annotated link|Càdlàg}}
* {{annotated link|Fatou's lemma}}

== Notes ==
{{reflist|group=note}}

== References ==
{{reflist|refs=
<ref name="freemanRobust">{{cite book | vauthors=((Freeman, R. A.)), ((Kokotović, P.)) | date= 1996 | title=Robust Nonlinear Control Design | publisher=Birkhäuser Boston | url=http://link.springer.com/10.1007/978-0-8176-4759-9 | doi=10.1007/978-0-8176-4759-9 | isbn=978-0-8176-4758-2}}.</ref>
<ref name="goebelSetvalued">
{{cite book | vauthors=((Goebel, R. K.)) | date= January 2024 | chapter=Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control: An Introduction | title=Chapter 2: Set convergence and set-valued mappings | publisher=Society for Industrial and Applied Mathematics | series=Other Titles in Applied Mathematics | pages=21–36 | url=https://epubs.siam.org/doi/10.1137/1.9781611977981.ch2 | doi=10.1137/1.9781611977981.ch2 | isbn=978-1-61197-797-4}}
</ref>
}}

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{{Convex analysis and variational analysis}}

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