Semi-continuity

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In mathematical analysis, semi-continuity (or semicontinuity) is a property of real-valued functions that is weaker than continuity. A real-valued function f is upper 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 f(x₀). If "less than" is replaced by "greater than", the function is called lower semi-continuous at x₀.

Examples

An upper semi-continuous function (the full blue ball indicates f(x₀)).

Consider the function 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.

A lower semi-continuous function (the full blue ball indicates f(x₀)).

The floor function [f(x)=\lfloor x \rfloor], which returns the greatest integer smaller than a given [x], is everywhere upper semi-continuous. Similarly the ceiling function [f(x)= \lceil x \rceil] is lower semi-continuous.

Formal definition

Suppose X is a topological space, x₀ is a point in X and f : X → R is a 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

lim sup _{x → x₀}f(x) ≤ f(x₀)

(see limit superior and limit inferior for the definition of lim sup). The function f is called upper semi-continuous if it is upper semi-continuous at every point of its domain. Then 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

lim inf _{x → x₀} f(x) ≥ f(x₀)

(see limit superior and limit inferior for the definition of lim inf). The function f is called lower semi-continuous if it is lower semi-continuous at every point of its domain. Then is an open set for every α∈R.

Properties

A function is continuous at x₀ if and only if it is upper and lower semi-continuous there.

If f and g are two functions which are both upper semi-continuous at x₀, then so is f + g. If both functions are non-negative, then the product function fg will also be upper semi-continuous at x₀. Multiplying a positive upper semi-continuous function with a negative number turns it into a lower semi-continuous function.

If C is a compact space (for instance a closed interval [a, b]) and f : C → R is upper semi-continuous, then f has a maximum on C. The analogous statement for lower semi-continuous functions and minima is also true.

Suppose f_n : X → R is a lower semi-continuous function for every natural number n, and

f(x) := sup < ∞

for every x in X. Then f is lower semi-continuous. Even if all the f_n are continuous, f need not be continuous.

The indicator function of any open set is lower semicontinuous. The characteristic function of a closed set is upper semicontinuous.

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