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 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 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 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