In mathematics, the concept of continuity as it is defined for single-valued functions is not immediately extendible to multi-valued mappings or correspondences. In order to derive a more generalized definition, the dual concepts of upper hemicontinuity and lower hemicontinuity are introduced. A correspondence that has both properties is said to be continuous in an analogy to the property of the same name for functions. Upper hemicontinuity is approximately when the graph of the correspondence is closed from the left and from the right (but not necessarily closed at every point). And Lower hemicontinuity is approximately when the graph has no closed edges (i.e. a point that a convergent sequence in the range can reach from the left or right, but not from the other direction).

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