Hemicontinuity

Summary

Hemicontinuity is the notion of continuity for multivalued functions, which assign to each point a set of values. Upper hemicontinuity says that the value set cannot suddenly grow, and lower hemicontinuity says that it cannot suddenly shrink. These notions appear in fixed-point theorems and in selection theorems such as the Michael selection theorem. They also appear in economics and game theory, where best responses are often set-valued.