Wikipedia
In mathematics, the notion of the continuity of functions is not immediately extensible to multi-valued 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 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.