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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