Hemicontinuity

In mathematics, the notion of the continuity of functions is not immediately extensible to set-valued functions between two sets A and B. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extension. A set-valued function that has both properties is said to be continuous in an analogy to the property of the same name for single-valued functions.

To explain both notions, consider a sequence a of points in a domain, and a sequence b of points in the range. We say that b corresponds to a if each point in b is contained in the image of the corresponding point in a.

• Upper hemicontinuity requires that, for any convergent sequence a in a domain, and for any convergent sequence b that corresponds to a, the image of the limit of a contains the limit of b.
• Lower hemicontinuity requires that, for any convergent sequence a in a domain, and for any point x in the image of the limit of a, there exists a sequence b that corresponds to a subsequence of a, that converges to x.

Examples

This set-valued function is upper hemicontinuous everywhere, but not lower hemicontinuous at $x$ : for a sequence of points $\left(x_{m}\right)$ that converges to $x,$ we have a $y$ ($y\in f(x)$) such that no sequence of $\left(y_{m}\right)$ converges to $y$ where each $y_{m}$ is in $f\left(x_{m}\right).$

This set-valued function is lower hemicontinuous everywhere, but not upper hemicontinuous at $x,$ because the graph (set) is not closed.

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).

Formal definition: upper hemicontinuity

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

Sequential characterization

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

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

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).

If B is compact, the converse is also true.

Closed graph theorem

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

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

Formal definition: lower hemicontinuity

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

Sequential characterization

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

Open graph theorem

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

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

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

Properties

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).

Implications for continuity

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.

Other concepts of continuity

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

$\Gamma :A\to B$ is lower [resp. upper] hemicontinuous if and only if the mapping $\Gamma :A\to P(B)$ 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).

See also

• 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

References

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• Aubin, Jean-Pierre; Cellina, Arrigo (1984). Differential Inclusions: Set-Valued Maps and Viability Theory. Grundl. der Math. Wiss. Vol. 264. Berlin: Springer. ISBN 0-387-13105-1.
• Aubin, Jean-Pierre; Frankowska, Hélène (1990). Set-Valued Analysis. Basel: Birkhäuser. ISBN 3-7643-3478-9.
• Deimling, Klaus (1992). Multivalued Differential Equations. Walter de Gruyter. ISBN 3-11-013212-5.
• Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995). Microeconomic Analysis. New York: Oxford University Press. pp. 949–951. ISBN 0-19-507340-1.
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