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 ${\displaystyle x}$ : for a sequence of points ${\displaystyle \left(x_{m}\right)}$ that converges to ${\displaystyle x,}$ we have a ${\displaystyle y}$ (${\displaystyle y\in f(x)}$) such that no sequence of ${\displaystyle \left(y_{m}\right)}$ converges to ${\displaystyle y}$ where each ${\displaystyle y_{m}}$ is in ${\displaystyle f\left(x_{m}\right).}$

This set-valued function is lower hemicontinuous everywhere, but not upper hemicontinuous at ${\displaystyle x,}$ 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 ${\displaystyle \Gamma :A\to B}$ 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

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

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]

Formal definition: lower hemicontinuity

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

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

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]

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:

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

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

Notes

1. Proposition 1.4.8 of Aubin, Jean-Pierre; Frankowska, Hélène (1990). Set-Valued Analysis. Basel: Birkhäuser. ISBN   3-7643-3478-9.
2. Zhou, J.X. (August 1995). "On the Existence of Equilibrium for Abstract Economies". Journal of Mathematical Analysis and Applications. 193 (3): 839–858. doi: 10.1006/jmaa.1995.1271 .

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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