Hemicontinuity

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

Roughly speaking, a function is upper hemicontinuous when (1) a convergent sequence of points in the domain maps to a sequence of sets in the range which (2) contain another convergent sequence, then the image of the limiting point in the domain must contain the limit of the sequence in the range. Lower hemicontinuity essentially reverses this, saying if a sequence in the domain converges, given a point in the range of the limit, then you can find a sub-sequence whose image contains a convergent sequence to the given point.

Upper hemicontinuity

This correspondence 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 ${\displaystyle f\left(x_{m}\right).}$

A correspondence ${\displaystyle \Gamma :A\to B}$ is said to be upper hemicontinuous at the point ${\displaystyle a}$ if for any open neighbourhood ${\displaystyle V}$ of ${\displaystyle \Gamma (a)}$ 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 correspondence ${\displaystyle \Gamma :A\to B}$ 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 ${\displaystyle A,}$ for all ${\displaystyle b\in B,}$ 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.

Closed graph theorem

The graph of a correspondence ${\displaystyle \Gamma :A\to B}$ 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 correspondence 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]

Lower hemicontinuity

This correspondence is lower hemicontinuous everywhere, but not upper hemicontinuous at ${\displaystyle x,}$ because the graph (set) is not closed.

A correspondence ${\displaystyle \Gamma :A\to B}$ 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 correspondence ${\displaystyle \Gamma :A\to B}$ 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 convex-valued correspondence with 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 multivalued maps (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 correspondences 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 multivalued maps. Typically lower hemicontinuous correspondences 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 correspondence 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
• Multivalued function – Generalization of a function that may produce several outputs for each input
• Semicontinuity

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

• Aliprantis, Charalambos D.; Border, Kim C. (2007). Infinite Dimensional Analysis: Hitchhiker's Guide (Third ed.). Berlin: Springer. ISBN 978-3-540-32696-0.
• Aubin, Jean-Pierre; Cellina, Arrigo (1984). Differential Inclusions: Set-Valued Maps and Viability Theory. Grundl. der Math. Wiss. 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.
• Ok, Efe A. (2007). Real Analysis with Economic Applications. Princeton University Press. pp. 216–226. ISBN 0-691-11768-3.