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.
|
