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Semicontinuity


In mathematical analysis, semi-continuity (or semicontinuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semi-continuous at a point x0 if, roughly speaking, the function values for arguments near x0 are either close to f(x0) or less than (respectively, greater than) f(x0).

Consider the function f, piecewise defined by f(x) = –1 for x < 0 and f(x) = 1 for x ≥ 0. This function is upper semi-continuous at x0 = 0, but not lower semi-continuous.

The indicator function of an open set is lower semi-continuous, whereas the indicator function of a closed set is upper semi-continuous. The floor function , which returns the greatest integer less than or equal to a given real number x, is everywhere upper semi-continuous. Similarly, the ceiling function is lower semi-continuous.


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