# Hemicontinuity

In mathematics, the notion of the continuity of functions is not immediately extensible to multi-valued mappings or correspondencesTemplate:Disambiguation needed. 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 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

A correspondence Γ : *A* → *B* is said to be **upper hemicontinuous** at the point *a* if for any open neighbourhood *V* of Γ(*a*) there exists a neighbourhood *U* of *a* such that for all *x* in *U*, Γ(*x*) is a subset of *V*.

### Closed Graph Theorem

For a correspondence Γ : *A* → *B* with closed values (i.e. Γ (*a*) - closed for *a* in *A*), closed domain and compact range, to be upper hemicontinuous it is sufficient and necessary to have closed graph. That is that the set:

### Sequential characterization

Γ : *A* → *B* is upper hemicontinuous at if , and

If Γ is compact-valued (i.e. Γ(*x*) is compact for all *x*) the converse is also true.

## Lower hemicontinuity

A correspondence Γ : *A* → *B* is said to be **lower hemicontinuous** at the point *a*
if for any open set *V* intersecting Γ(*a*) there exists neighbourhood *U* of *a* such that Γ(*x*) intersects *V* for all *x* in *U*. (Here *V* *intersects* *S* means nonempty intersection ).

### Sequential characterization

Γ : *A* → *B* is lower hemicontinuous at *a* if and only if

### Open Graph Theorem

If Γ : *A* → *B* has open graph *Gr*(Γ), then it is lower hemicontinuous.

## 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.

Very important part of set-valued analysis (in view of applications) constitutes 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:

- Γ :
*A*→*B*is lower [resp. upper] hemicontinuous if and only if the mapping Γ :*A*→*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

## References

- Jean-Pierre Aubin, Arrigo Cellina
*Differential Inclusions, Set-Valued Maps And Viability Theory*, Grundl. der Math. Wiss., vol. 264, Springer - Verlag, Berlin, 1984 - Jean-Pierre Aubin, Helene Frankowska
*Set-Valued Analysis*, Birkh¨auser, Basel, 1990 - Klaus Deimling
*Multivalued Differential Equations*, Walter de Gruyter, 1992 - Charalambos D. Aliprantis, Kim C. Border
*Infinite dimensional analysis. Hitchhiker's guide*, Springer, 1994(?) - Mas-Colell, Whinston, and Green.
*Microeconomic Analysis*, Oxford University Press, 1995, pp 949–951. - {{#invoke:citation/CS1|citation

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