Editing Semi-continuity (section)

== 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. 
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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.
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=== 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.-->