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.
A correspondence is said to be upper hemicontinuous at the point if for any open neighbourhood of there exists a neighbourhood of such that for all is a subset of
For a correspondence with closed values, if is upper hemicontinuous at then for all sequences in for all all sequences such that
If B is compact, the converse is also true.
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.
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 intersects means nonempty intersection ).
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
A correspondence 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.
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.
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).
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.
The upper and lower hemicontinuity might be viewed as usual continuity:
(For the notion of hyperspace compare also power set and function space).
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