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