Semicontinuity

Summary

Semicontinuity is a one-sided form of continuity for functions with values in the extended real line. A function is upper semicontinuous when its value at a point may lie above the nearby values but not below them. It is lower semicontinuous in the opposite case. A function is continuous exactly when it is both. Lower semicontinuous functions attain their minimum on nonempty compact spaces, which makes them basic in optimisation and the calculus of variations.