Semicontinuity

In mathematical analysis, semicontinuity is a property of real-valued functions that is weaker than continuity. It comes in two kinds, upper semicontinuity and lower semicontinuity.

A real-valued function over a topological space

is said to be lower semicontinuous if the following property holds:

is an open set for every .

It is said to be an upper semicontinuous function if the following property holds:

is an open set for every .

Properties

• A function is continuous if and only if it is both upper and lower semicontinuous.
• The characteristic function of an open set is lower semicontinuous.
• The characteristic function of a closed set is upper semicontinuous.