Upper hemicontinuity
A correspondence is said to be upper hemicontinuous at the point if, for any open with , there exists a neighbourhood of such that for all is a subset ofSequential characterization
For a correspondence with closed values, if is upper hemicontinuous at then for all sequences in for all all sequences such that :if and then If B is compact, the converse is also true.Closed graph theorem
The graph of a correspondence is the set defined by If is an upper hemicontinuous correspondence with closed domain (that is, the set of points where is not the empty set is closed) and closed values (i.e. is closed for all ), then is closed. If is compact, then the converse is also true.Proposition 1.4.8 ofLower hemicontinuity
A correspondence is said to be lower hemicontinuous at the point if for any open set intersecting there exists a neighbourhood of such that intersects for all (Here means nonempty intersection ).Sequential characterization
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 everyOpen graph theorem
A correspondence have if the set is open in for every If values are all open sets in then is said to have . If has an open graph then has open upper and lower sections and if has open lower sections then it is lower hemicontinuous. The open graph theorem says that if is a convex-valued correspondence with open upper sections, then has an open graph in if and only if 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. 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: : is lower esp. upperhemicontinuous if and only if the mapping is continuous where theSee also
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References
* * * * * * {{Convex analysis and variational analysis Theory of continuous functions Mathematical analysis Variational analysis