In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, roughly speaking, the function values for arguments near x_0 are not much higher (respectively, lower) than A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point x_0 to for some c>0, then the result is upper semicontinuous; if we decrease its value to then the result is lower semicontinuous. The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899.

Definitions

Assume throughout that X is a topological space and is a function with values in the extended real numbers.

Upper semicontinuity

A function is called upper semicontinuous at a point x_0 \in X if for every real there exists a neighborhood U of x_0 such that f(x)<y for all x\in U. Equivalently, f is upper semicontinuous at x_0 if and only if where lim sup is the limit superior of the function f at the point x_0. If X is a metric space with distance function d and this can also be restated using an \varepsilon-\delta formulation, similar to the definition of continuous function. Namely, for each there is a \delta>0 such that whenever A function is called upper semicontinuous if it satisfies any of the following equivalent conditions:

Lower semicontinuity

A function is called lower semicontinuous at a point x_0\in X if for every real there exists a neighborhood U of x_0 such that f(x)>y for all x\in U. Equivalently, f is lower semicontinuous at x_0 if and only if where \liminf is the limit inferior of the function f at point x_0. If X is a metric space with distance function d and this can also be restated as follows: For each there is a \delta>0 such that whenever A function is called lower semicontinuous if it satisfies any of the following equivalent conditions:

Examples

Consider the function f, piecewise defined by: This function is upper semicontinuous at x_0 = 0, but not lower 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. Upper and lower semicontinuity bear no relation to continuity from the left or from the right for functions of a real variable. Semicontinuity is defined in terms of an ordering in the range of the functions, not in the domain. For example the function is upper semicontinuous at x = 0 while the function limits from the left or right at zero do not even exist. If X = \R^n is a Euclidean space (or more generally, a metric space) and is the space of curves in X (with the supremum distance ), then the length functional which assigns to each curve \alpha its length L(\alpha), is lower semicontinuous. As an example, consider approximating the unit square diagonal by a staircase from below. The staircase always has length 2, while the diagonal line has only length \sqrt 2. Let (X,\mu) be a measure space and let L^+(X,\mu) denote the set of positive measurable functions endowed with the topology of convergence in measure with respect to \mu. Then by Fatou's lemma the integral, seen as an operator from L^+(X,\mu) to is lower semicontinuous. Tonelli's theorem in functional analysis characterizes the weak lower semicontinuity of nonlinear functionals on Lp spaces in terms of the convexity of another function.

Properties

Unless specified otherwise, all functions below are from a topological space X to the extended real numbers Several of the results hold for semicontinuity at a specific point, but for brevity they are only stated for semicontinuity over the whole domain.

Binary Operations on Semicontinuous Functions

Let.

Optimization of Semicontinuous Functions

Other Properties

Semicontinuity of Set-valued Functions

For set-valued functions, several concepts of semicontinuity have been defined, namely upper, lower, outer, and inner semicontinuity, as well as upper and lower hemicontinuity. A set-valued function F from a set A to a set B is written For each x \in A, the function F defines a set The preimage of a set S \subset B under F is defined as That is, F^{-1}(S) is the set that contains every point x in A such that F(x) is not disjoint from S.

Upper and Lower Semicontinuity

A set-valued map is upper semicontinuous at if for every open set such that, there exists a neighborhood V of x such that A set-valued map is lower semicontinuous at if for every open set such that there exists a neighborhood V of x such that Upper and lower set-valued semicontinuity are also defined more generally for a set-valued maps between topological spaces by replacing and in the above definitions with arbitrary topological spaces. Note, that there is not a direct correspondence between single-valued lower and upper semicontinuity and set-valued lower and upper semicontinuouty. An upper semicontinuous single-valued function is not necessarily upper semicontinuous when considered as a set-valued map. For example, the function defined by is upper semicontinuous in the single-valued sense but the set-valued map is not upper semicontinuous in the set-valued sense.

Inner and Outer Semicontinuity

A set-valued function is called inner semicontinuous at x if for every y \in F(x) and every convergent sequence (x_i) in such that x_i \to x, there exists a sequence (y_i) in such that y_i \to y and for all sufficiently large A set-valued function is called outer semicontinuous at x if for every convergence sequence (x_i) in such that x_i \to x and every convergent sequence (y_i) in such that for each the sequence (y_i) converges to a point in F(x) (that is, ).