Hemicontinuity
In mathematics, the notion of the continuity of functions is not immediately extensible to set-valued functions between two sets A and B. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extension. A set-valued function that has both properties is said to be continuous in an analogy to the property of the same name for single-valued functions. Roughly speaking, a function is upper hemicontinuous if 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. Upper hemicontinuity A set-valued function 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 of Sequential characterization For a set-valued function with closed values, if is upper hemicontinuous at then for all sequences in and all sequences such that if and then If B is compact, the converse is also true. Closed graph theorem The graph of a set-valued function is the set defined by If is an upper hemicontinuous set-valued function 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. Lower hemicontinuity A set-valued function 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 every Open graph theorem A set-valued function 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 set-valued function with convex values and open upper sections, then has an open graph in if and only if is lower hemicontinuous. Properties 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