In mathematics, upper hemicontinuity and lower hemicontinuity are extensions of the notions of upper and lower semicontinuity of single-valued functions to set-valued functions. A set-valued function that is both upper and lower hemicontinuous 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

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This set-valued function is upper hemicontinuous everywhere, but not lower hemicontinuous at   : for a sequence of points   that converges to   we have a   ( ) such that no sequence of   converges to   where each   is in  
 
This set-valued function is lower hemicontinuous everywhere, but not upper hemicontinuous at   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).

Definitions

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Upper hemicontinuity

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A set-valued function   is said to be upper hemicontinuous at a point   if, for every open   with   there exists a neighbourhood   of   such that for all     is a subset of  

Lower hemicontinuity

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A set-valued function   is said to be lower hemicontinuous at the point   if for every open set   intersecting   there exists a neighbourhood   of   such that   intersects   for all   (Here   intersects   means nonempty intersection  ).

Continuity

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

Properties

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Upper hemicontinuity

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Sequential characterization

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Theorem — For a set-valued function   with closed values, if   is upper hemicontinuous at   then for every sequence   in   and every sequence   such that  

if   and   then  

If   is compact, then the converse is also true.

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).

Closed graph theorem

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The graph of a set-valued function   is the set defined by   The graph of   is the set of all   such that   is not empty.

Theorem — If   is an upper hemicontinuous set-valued function with closed domain (that is, the domain of   is closed) and closed values (i.e.   is closed for all  ), then   is closed.

If   is compact, then the converse is also true.[1]

Lower hemicontinuity

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Sequential characterization

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Theorem —   is lower hemicontinuous at   if and only if for every sequence   in   such that   in   and all   there exists a subsequence   of   and also a sequence   such that   and   for every  

Open graph theorem

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A set-valued function   is said to have open lower sections if the set   is open in   for every   If   values are all open sets in   then   is said to have open upper sections.

If   has an open graph   then   has open upper and lower sections and if   has open lower sections then it is lower hemicontinuous.[2]

Open Graph Theorem — If   is a set-valued function with convex values and open upper sections, then   has an open graph in   if and only if   is lower hemicontinuous.[2]

Operations Preserving Hemicontinuity

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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.

Function Selections

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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).

Other concepts of continuity

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

Theorem —  A set-valued map   is lower [resp. upper] hemicontinuous if and only if the mapping   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

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Notes

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  1. ^ Proposition 1.4.8 of Aubin, Jean-Pierre; Frankowska, Hélène (1990). Set-Valued Analysis. Basel: Birkhäuser. ISBN 3-7643-3478-9.
  2. ^ a b 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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