The image on the right shows a function that is not lower hemicontinuous at x. To see this, let a be a sequence that converges to x from the left. The image of x is a vertical line that contains some point (x,y). But every sequence b that corresponds to a is contained in the bottom horizontal line, so it cannot converge to y. In contrast, the function is upper hemicontinuous everywhere. For example, considering any sequence a that converges to x from the left or from the right, and any corresponding sequence b, the limit of b is contained in the vertical line that is the image of the limit of a.

The image on the left shows a function that is not upper hemicontinuous at x. To see this, let a be a sequence that converges to x from the right. The image of a contains vertical lines, so there exists a corresponding sequence b in which all elements are bounded away from f(x). The image of the limit of a contains a single point f(x), so it does not contain the limit of b. In contrast, that function is lower hemicontinuous everywhere. For example, for any sequence a that converges to x, from the left or from the right, f(x) contains a single point, and there exists a corresponding sequence b that converges to f(x).

A set-valued function ${\displaystyle \Gamma :A\to B}$  is said to be upper hemicontinuous at the point ${\displaystyle a}$  if, for any open ${\displaystyle V\subset B}$  with ${\displaystyle \Gamma (a)\subset V}$ , there exists a neighbourhood ${\displaystyle U}$  of ${\displaystyle a}$  such that for all ${\displaystyle x\in U,}$  ${\displaystyle \Gamma (x)}$  is a subset of ${\displaystyle V.}$ 

Sequential characterization edit

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 ${\displaystyle b\in \Gamma (a).}$ 

As an example, look at the image at the right, and consider sequence a in the domain that converges to x (either from the left or from the right). Then, any sequence b that satisfies the requirements converges to some point in f(x). Therefore, the

If B is compact, the converse is also true.

Closed graph theorem edit

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]

A set-valued function ${\displaystyle \Gamma :A\to B}$  is said to be lower hemicontinuous at the point ${\displaystyle a}$  if for any open set ${\displaystyle V}$  intersecting ${\displaystyle \Gamma (a)}$  there exists a neighbourhood ${\displaystyle U}$  of ${\displaystyle a}$  such that ${\displaystyle \Gamma (x)}$  intersects ${\displaystyle V}$  for all ${\displaystyle x\in U.}$  (Here ${\displaystyle V}$  intersects ${\displaystyle S}$  means nonempty intersection ${\displaystyle V\cap S\neq \varnothing }$ ).

Sequential characterization edit

${\displaystyle \Gamma :A\to B}$  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 ${\displaystyle k.}$ 

Open graph theorem edit

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]

Set-theoretic, algebraic and topological operations on set-valued functions (like union, composition, sum, convex hull, closure) usually preserve the type of continuity. But this should be taken with appropriate care since, for example, there exists a pair of lower hemicontinuous set-valued functions whose intersection is not lower hemicontinuous. This can be fixed upon strengthening continuity properties: if one of those lower hemicontinuous multifunctions has open graph then their intersection is again lower hemicontinuous.

Crucial to set-valued analysis (in view of applications) are the investigation of single-valued selections and approximations to set-valued functions. Typically lower hemicontinuous set-valued functions admit single-valued selections (Michael selection theorem, Bressan–Colombo directionally continuous selection theorem, Fryszkowski decomposable map selection). Likewise, upper hemicontinuous maps admit approximations (e.g. Ancel–Granas–Górniewicz–Kryszewski theorem).

If a set-valued function is both upper hemicontinuous and lower hemicontinuous, it is said to be continuous. A continuous function is in all cases both upper and lower hemicontinuous.

The upper and lower hemicontinuity might be viewed as usual continuity:

${\displaystyle \Gamma :A\to B}$  is lower [resp. upper] hemicontinuous if and only if the mapping ${\displaystyle \Gamma :A\to P(B)}$  is continuous where the hyperspace P(B) has been endowed with the lower [resp. upper] Vietoris topology.

(For the notion of hyperspace compare also power set and function space).

Using lower and upper Hausdorff uniformity we can also define the so-called upper and lower semicontinuous maps in the sense of Hausdorff (also known as metrically lower / upper semicontinuous maps).

• Differential inclusion
• Hausdorff distance – Distance between two metric-space subsets
• Semicontinuity – Property of functions which is weaker than continuityPages displaying short descriptions of redirect targets

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