# 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