In mathematical analysis, semicontinuity (or semicontinuity) is a property of extended realvalued functions that is weaker than continuity. An extended realvalued function f is upper (lower) semicontinuous at a point x_{0} if, roughly speaking, the function values for arguments near x_{0} are either close to f(x_{0}) or less than (greater than) f(x_{0}).
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 semicontinuous at x_{0} = 0, but not lower semicontinuous.
The indicator function of an open set is lower semicontinuous, whereas the indicator function of a closed set is upper semicontinuous. The floor function , which returns the greatest integer less than or equal to a given real number x, is everywhere upper semicontinuous. Similarly, the ceiling function is lower semicontinuous.
A function may be upper or lower semicontinuous without being either left or right continuous. For example, the function
is upper semicontinuous 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 semicontinuous 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, x_{0} is a point in X and f : X → R ∪ {–∞,+∞} is an extended realvalued function. We say that f is upper semicontinuous at x_{0} if for every ε > 0 there exists a neighborhood U of x_{0} such that f(x) ≤ f(x_{0}) + 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 x_{0}).
The function f is called upper semicontinuous if it is upper semicontinuous at every point of its domain. A function is upper semicontinuous if and only if {x ∈ X : f(x) < α} is an open set for every α ∈ R.
We say that f is lower semicontinuous at x_{0} if for every ε > 0 there exists a neighborhood U of x_{0} such that f(x) ≥ f(x_{0}) – ε for all x in U. Equivalently, this can be expressed as
Full article ▸
