Hemicontinuity

For a set-valued function ${\displaystyle \Gamma :A\to B}$ with closed values, if ${\displaystyle \Gamma :A\to B}$ is upper hemicontinuous at ${\displaystyle a\in A}$ then for all sequences ${\displaystyle a_{\bullet }=\left(a_{m}\right)_{m=1}^{\infty }}$ in ${\displaystyle A}$ and all sequences ${\displaystyle \left(b_{m}\right)_{m=1}^{\infty }}$ such that ${\displaystyle b_{m}\in \Gamma \left(a_{m}\right),}$

if ${\displaystyle \lim _{m\to \infty }a_{m}=a}$ and ${\displaystyle \lim _{m\to \infty }b_{m}=b}$ then ${\displaystyle b\in \Gamma (a).}$

If B is compact, the converse is also true.

The graph of a set-valued function ${\displaystyle \Gamma :A\to B}$ is the set defined by ${\displaystyle Gr(\Gamma )=\{(a,b)\in A\times B:b\in \Gamma (a)\}.}$

If ${\displaystyle \Gamma :A\to B}$ is an upper hemicontinuous set-valued function with closed domain (that is, the set of points ${\displaystyle a\in A}$ where ${\displaystyle \Gamma (a)}$ is not the empty set is closed) and closed values (i.e. ${\displaystyle \Gamma (a)}$ is closed for all ${\displaystyle a\in A}$), then ${\displaystyle \operatorname {Gr} (\Gamma )}$ is closed. If ${\displaystyle B}$ is compact, then the converse is also true.[1]

${\displaystyle \Gamma :A\to B}$ is lower hemicontinuous at ${\displaystyle a}$ if and only if for every sequence ${\displaystyle a_{\bullet }=\left(a_{m}\right)_{m=1}^{\infty }}$ in ${\displaystyle A}$ such that ${\displaystyle a_{\bullet }\to a}$ in ${\displaystyle A}$ and all ${\displaystyle b\in \Gamma (a),}$ there exists a subsequence ${\displaystyle \left(a_{m_{k}}\right)_{k=1}^{\infty }}$ of ${\displaystyle a_{\bullet }}$ and also a sequence ${\displaystyle b_{\bullet }=\left(b_{k}\right)_{k=1}^{\infty }}$ such that ${\displaystyle b_{\bullet }\to b}$ and ${\displaystyle b_{k}\in \Gamma \left(a_{m_{k}}\right)}$ for every ${\displaystyle k.}$

A set-valued function ${\displaystyle \Gamma :A\to B}$ have open lower sections if the set ${\displaystyle \Gamma ^{-1}(b)=\{a\in A:b\in \Gamma (a)\}}$ is open in ${\displaystyle A}$ for every ${\displaystyle b\in B.}$ If ${\displaystyle \Gamma }$ values are all open sets in ${\displaystyle B,}$ then ${\displaystyle \Gamma }$ is said to have open upper sections.

If ${\displaystyle \Gamma }$ has an open graph ${\displaystyle \operatorname {Gr} (\Gamma ),}$ then ${\displaystyle \Gamma }$ has open upper and lower sections and if ${\displaystyle \Gamma }$ has open lower sections then it is lower hemicontinuous.[2]

The open graph theorem says that if ${\displaystyle \Gamma :A\to P\left(\mathbb {R} ^{n}\right)}$ is a set-valued function with convex values and open upper sections, then ${\displaystyle \Gamma }$ has an open graph in ${\displaystyle A\times \mathbb {R} ^{n}}$ if and only if ${\displaystyle \Gamma }$ is lower hemicontinuous.[2]