Semi-continuity

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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 (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 (greater than) f(x0).

Contents

Examples

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 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 f(x)=\lceil x \rceil is lower semi-continuous.

A function may be upper or lower semi-continuous without being either left or right continuous. For example, the function

is upper semi-continuous at x = 1 although not left or right continuous. The limit from the left is equal to 1 and the limit from the right is equal to 1/2, both of which are different from the function value of 2. Similarly the function

is upper semi-continuous at x = 0 while the function limits from the left or right at zero do not even exist.

Formal definition

Suppose X is a topological space, x0 is a point in X and f : X → R ∪ {–∞,+∞} is an extended real-valued function. We say that f is upper semi-continuous at x0 if for every ε > 0 there exists a neighborhood U of x0 such that f(x) ≤ f(x0) + ε for all x in U. Equivalently, this can be expressed as

where lim sup is the limit superior (of the function f at point x0).

The function f is called upper semi-continuous if it is upper semi-continuous at every point of its domain. A function is upper semi-continuous if and only if {x ∈ X : f(x) < α} is an open set for every α ∈ R.

We say that f is lower semi-continuous at x0 if for every ε > 0 there exists a neighborhood U of x0 such that f(x) ≥ f(x0) – ε for all x in U. Equivalently, this can be expressed as

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