Hemicontinuity

In mathematics, the concept of continuity as it is defined for single-valued functions is not immediately extendible to multi-valued mappings or correspondences. In order to derive a more generalized definition, the dual concepts of upper hemicontinuity and lower hemicontinuity are introduced. A correspondence that has both properties is said to be continuous in an analogy to the property of the same name for functions.

Upper hemicontinuity is approximately when the graph of the correspondence is closed from the left and from the right (but not necessarily closed at every point). And Lower hemicontinuity is approximately when the graph has no closed edges (i.e. a point that a convergent sequence in the range can reach from the left or right, but not from the other direction).

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 neighbourhood U of a s.t. Γ(x) is in V for all x in U.

One should be warned that the term upper semicontinuous, instead of upper hemicontinuous, has been more widespreaded in the literature. Then upper hemicontinuous is reserved for upper semicontinuity w.r.t. weak topology.

Multivalued Closed Graph Theorem: For correspondence Γ : A → B with closed values (i.e. Γ (a) - closed for a in A) and compact range (i.e. $Gamma(A) = bigcup_{ain A} Gamma(a)$ is compact subset of B) to be upper hemicontinuous it is sufficient and necessary to have closed graph $Gr(Gamma) = {(a,b)in Atimes B : binGamma(a)}$ in A×B; sequentially:

forall a, a_m in A, forall b, b_m in B, , a_m rarr a, Gamma(a_m) ni b_m rarr b , implies b , in Gamma(a)

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 s.t. Γ(x) intersects V for all x in U. (Here V intersects S means nonempty intersection $V cap S neqemptyset$ ).

One should be warned that the term lower semicontinuous, instead of lower hemicontinuous, has been more widespreaded in the literature. Then lower hemicontinuous is reserved for lower semicontinuity w.r.t. weak topology.

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

forall a, a_m in A, , a_m rarr a, forall b in Gamma(a), exists a_{m_k}

subsequence of

a_m, , exists b_k in Gamma(a_{m_k}), , b_k rarr b

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).

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
Ch.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.