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 ${\displaystyle f(x)=\lfloor x\rfloor }$, which returns the greatest integer less than or equal to a given real number x, is everywhere upper semi-continuous. Similarly, the ceiling function ${\displaystyle f(x)=\lceil x\rceil }$ is lower semi-continuous.

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