Semi-continuity

Short description: Property of functions which is weaker than continuity

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 [math]\displaystyle{ f }[/math] is upper (respectively, lower) semicontinuous at a point [math]\displaystyle{ x_0 }[/math] if, roughly speaking, the function values for arguments near [math]\displaystyle{ x_0 }[/math] are not much higher (respectively, lower) than [math]\displaystyle{ f\left(x_0\right). }[/math]

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 [math]\displaystyle{ x_0 }[/math] to [math]\displaystyle{ f\left(x_0\right) + c }[/math] for some [math]\displaystyle{ c\gt 0 }[/math], then the result is upper semicontinuous; if we decrease its value to [math]\displaystyle{ f\left(x_0\right) - c }[/math] then the result is lower semicontinuous.

An upper semicontinuous function that is not lower semicontinuous. The solid blue dot indicates [math]\displaystyle{ f\left(x_0\right). }[/math]

A lower semicontinuous function that is not upper semicontinuous. The solid blue dot indicates [math]\displaystyle{ f\left(x_0\right). }[/math]

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 [math]\displaystyle{ X }[/math] is a topological space and [math]\displaystyle{ f:X\to\overline{\R} }[/math] is a function with values in the extended real numbers [math]\displaystyle{ \overline{\R}=\R \cup \{-\infty,\infty\} = [-\infty,\infty] }[/math].

Upper semicontinuity

A function [math]\displaystyle{ f:X\to\overline{\R} }[/math] is called upper semicontinuous at a point [math]\displaystyle{ x_0 \in X }[/math] if for every real [math]\displaystyle{ y \gt f\left(x_0\right) }[/math] there exists a neighborhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ x_0 }[/math] such that [math]\displaystyle{ f(x)\lt y }[/math] for all [math]\displaystyle{ x\in U }[/math]. Equivalently, [math]\displaystyle{ f }[/math] is upper semicontinuous at [math]\displaystyle{ x_0 }[/math] if and only if [math]\displaystyle{ \limsup_{x \to x_0} f(x) \leq f(x_0) }[/math] where lim sup is the limit superior of the function [math]\displaystyle{ f }[/math] at the point [math]\displaystyle{ x_0 }[/math].

A function [math]\displaystyle{ f:X\to\overline{\R} }[/math] is called upper semicontinuous if it satisfies any of the following equivalent conditions:

(1) The function is upper semicontinuous at every point of its domain.

(2) All sets [math]\displaystyle{ f^{-1}([ -\infty ,y))=\{x\in X : f(x)\lt y\} }[/math] with [math]\displaystyle{ y\in\R }[/math] are open in [math]\displaystyle{ X }[/math], where [math]\displaystyle{ [ -\infty ,y)=\{t\in\overline{\R}:t\lt y\} }[/math].

(3) All superlevel sets [math]\displaystyle{ \{x\in X : f(x)\ge y\} }[/math] with [math]\displaystyle{ y\in\R }[/math] are closed in [math]\displaystyle{ X }[/math].

(4) The hypograph [math]\displaystyle{ \{(x,t)\in X\times\R : t\le f(x)\} }[/math] is closed in [math]\displaystyle{ X\times\R }[/math].

(5) The function is continuous when the codomain [math]\displaystyle{ \overline{\R} }[/math] is given the left order topology. This is just a restatement of condition (2) since the left order topology is generated by all the intervals [math]\displaystyle{ [ -\infty,y) }[/math].

Lower semicontinuity

A function [math]\displaystyle{ f:X\to\overline{\R} }[/math] is called lower semicontinuous at a point [math]\displaystyle{ x_0\in X }[/math] if for every real [math]\displaystyle{ y \lt f\left(x_0\right) }[/math] there exists a neighborhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ x_0 }[/math] such that [math]\displaystyle{ f(x)\gt y }[/math] for all [math]\displaystyle{ x\in U }[/math]. Equivalently, [math]\displaystyle{ f }[/math] is lower semicontinuous at [math]\displaystyle{ x_0 }[/math] if and only if [math]\displaystyle{ \liminf_{x \to x_0} f(x) \ge f(x_0) }[/math] where [math]\displaystyle{ \liminf }[/math] is the limit inferior of the function [math]\displaystyle{ f }[/math] at point [math]\displaystyle{ x_0 }[/math].

A function [math]\displaystyle{ f:X\to\overline{\R} }[/math] is called lower semicontinuous if it satisfies any of the following equivalent conditions:

(1) The function is lower semicontinuous at every point of its domain.

(2) All sets [math]\displaystyle{ f^{-1}((y,\infty ])=\{x\in X : f(x)\gt y\} }[/math] with [math]\displaystyle{ y\in\R }[/math] are open in [math]\displaystyle{ X }[/math], where [math]\displaystyle{ (y,\infty ]=\{t\in\overline{\R}:t\gt y\} }[/math].

(3) All sublevel sets [math]\displaystyle{ \{x\in X : f(x)\le y\} }[/math] with [math]\displaystyle{ y\in\R }[/math] are closed in [math]\displaystyle{ X }[/math].

(4) The epigraph [math]\displaystyle{ \{(x,t)\in X\times\R : t\ge f(x)\} }[/math] is closed in [math]\displaystyle{ X\times\R }[/math].

(5) The function is continuous when the codomain [math]\displaystyle{ \overline{\R} }[/math] is given the right order topology. This is just a restatement of condition (2) since the right order topology is generated by all the intervals [math]\displaystyle{ (y,\infty ] }[/math].

Examples

Consider the function [math]\displaystyle{ f, }[/math] piecewise defined by: [math]\displaystyle{ f(x) = \begin{cases} -1 & \mbox{if } x \lt 0,\\ 1 & \mbox{if } x \geq 0 \end{cases} }[/math] This function is upper semicontinuous at [math]\displaystyle{ x_0 = 0, }[/math] but not lower semicontinuous.

The floor function [math]\displaystyle{ f(x) = \lfloor x \rfloor, }[/math] which returns the greatest integer less than or equal to a given real number [math]\displaystyle{ x, }[/math] is everywhere upper semicontinuous. Similarly, the ceiling function [math]\displaystyle{ f(x) = \lceil x \rceil }[/math] 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 [math]\displaystyle{ f(x) = \begin{cases} \sin(1/x) & \mbox{if } x \neq 0,\\ 1 & \mbox{if } x = 0, \end{cases} }[/math] is upper semicontinuous at [math]\displaystyle{ x = 0 }[/math] while the function limits from the left or right at zero do not even exist.

If [math]\displaystyle{ X = \R^n }[/math] is a Euclidean space (or more generally, a metric space) and [math]\displaystyle{ \Gamma = C([0,1], X) }[/math] is the space of curves in [math]\displaystyle{ X }[/math] (with the supremum distance [math]\displaystyle{ d_\Gamma(\alpha,\beta) = \sup\{d_X(\alpha(t),\beta(t)):t\in[0,1]\} }[/math]), then the length functional [math]\displaystyle{ L : \Gamma \to [0, +\infty], }[/math] which assigns to each curve [math]\displaystyle{ \alpha }[/math] its length [math]\displaystyle{ L(\alpha), }[/math] 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 [math]\displaystyle{ \sqrt 2 }[/math].

Let [math]\displaystyle{ (X,\mu) }[/math] be a measure space and let [math]\displaystyle{ L^+(X,\mu) }[/math] denote the set of positive measurable functions endowed with the topology of convergence in measure with respect to [math]\displaystyle{ \mu. }[/math] Then by Fatou's lemma the integral, seen as an operator from [math]\displaystyle{ L^+(X,\mu) }[/math] to [math]\displaystyle{ [-\infty, +\infty] }[/math] is lower semicontinuous.

Properties

Unless specified otherwise, all functions below are from a topological space [math]\displaystyle{ X }[/math] to the extended real numbers [math]\displaystyle{ \overline{\R}= [-\infty,\infty]. }[/math] Several of the results hold for semicontinuity at a specific point, but for brevity they are only stated from semicontinuity over the whole domain.

• A function [math]\displaystyle{ f:X\to\overline{\R} }[/math] is continuous if and only if it is both upper and lower semicontinuous.

• The indicator function of a set [math]\displaystyle{ A\subset X }[/math] (defined by [math]\displaystyle{ \mathbf{1}_A(x)=1 }[/math] if [math]\displaystyle{ x\in A }[/math] and [math]\displaystyle{ 0 }[/math] if [math]\displaystyle{ x\notin A }[/math]) is upper semicontinuous if and only if [math]\displaystyle{ A }[/math] is a closed set. It is lower semicontinuous if and only if [math]\displaystyle{ A }[/math] is an open set.

• The sum [math]\displaystyle{ f+g }[/math] of two lower semicontinuous functions is lower semicontinuous (provided the sum is well-defined, i.e., [math]\displaystyle{ f(x)+g(x) }[/math] is not the indeterminate form [math]\displaystyle{ -\infty+\infty }[/math]). The same holds for upper semicontinuous functions.

• If both functions are non-negative, the product function [math]\displaystyle{ f g }[/math] of two lower semicontinuous functions is lower semicontinuous. The corresponding result holds for upper semicontinuous functions.

• A function [math]\displaystyle{ f:X\to\overline{\R} }[/math] is lower semicontinuous if and only if [math]\displaystyle{ -f }[/math] is upper semicontinuous.

• The composition [math]\displaystyle{ f \circ g }[/math] of upper semicontinuous functions is not necessarily upper semicontinuous, but if [math]\displaystyle{ f }[/math] is also non-decreasing, then [math]\displaystyle{ f \circ g }[/math] is upper semicontinuous.

• The minimum and the maximum of two lower semicontinuous functions are lower semicontinuous. In other words, the set of all lower semicontinuous functions from [math]\displaystyle{ X }[/math] to [math]\displaystyle{ \overline{\R} }[/math] (or to [math]\displaystyle{ \R }[/math]) forms a lattice. The same holds for upper semicontinuous functions.

• The (pointwise) supremum of an arbitrary family [math]\displaystyle{ (f_i)_{i\in I} }[/math] of lower semicontinuous functions [math]\displaystyle{ f_i:X\to\overline{\R} }[/math] (defined by [math]\displaystyle{ f(x)=\sup\{f_i(x):i\in I\} }[/math]) is lower semicontinuous.

In particular, the limit of a monotone increasing sequence [math]\displaystyle{ f_1\le f_2\le f_3\le\cdots }[/math] of continuous functions is lower semicontinuous. (The Theorem of Baire below provides a partial converse.) The limit function will only be lower semicontinuous in general, not continuous. An example is given by the functions [math]\displaystyle{ f_n(x)=1-(1-x)^n }[/math] defined for [math]\displaystyle{ x\in[0,1] }[/math] for [math]\displaystyle{ n=1,2,\ldots. }[/math]

Likewise, the infimum of an arbitrary family of upper semicontinuous functions is upper semicontinuous. And the limit of a monotone decreasing sequence of continuous functions is upper semicontinuous.

• (Theorem of Baire) Assume [math]\displaystyle{ X }[/math] is a metric space. Every lower semicontinuous function [math]\displaystyle{ f:X\to\overline{\R} }[/math] is the limit of a monotone increasing sequence of extended real-valued continuous functions on [math]\displaystyle{ X }[/math]; if [math]\displaystyle{ f }[/math] does not take the value [math]\displaystyle{ -\infty }[/math], the continuous functions can be taken to be real-valued.

And every upper semicontinuous function [math]\displaystyle{ f:X\to\overline{\R} }[/math] is the limit of a monotone decreasing sequence of extended real-valued continuous functions on [math]\displaystyle{ X }[/math]; if [math]\displaystyle{ f }[/math] does not take the value [math]\displaystyle{ \infty, }[/math] the continuous functions can be taken to be real-valued.

• If [math]\displaystyle{ C }[/math] is a compact space (for instance a closed bounded interval [math]\displaystyle{ [a, b] }[/math]) and [math]\displaystyle{ f : C \to \overline{\R} }[/math] is upper semicontinuous, then [math]\displaystyle{ f }[/math] has a maximum on [math]\displaystyle{ C. }[/math] If [math]\displaystyle{ f }[/math] is lower semicontinuous on [math]\displaystyle{ C, }[/math] it has a minimum on [math]\displaystyle{ C. }[/math]

(Proof for the upper semicontinuous case: By condition (5) in the definition, [math]\displaystyle{ f }[/math] is continuous when [math]\displaystyle{ \overline{\R} }[/math] is given the left order topology. So its image [math]\displaystyle{ f(C) }[/math] is compact in that topology. And the compact sets in that topology are exactly the sets with a maximum. For an alternative proof, see the article on the extreme value theorem.)

• Any upper semicontinuous function [math]\displaystyle{ f : X \to \N }[/math] on an arbitrary topological space [math]\displaystyle{ X }[/math] is locally constant on some dense open subset of [math]\displaystyle{ X. }[/math]

• Tonelli's theorem in functional analysis characterizes the weak lower semicontinuity of nonlinear functionals on L^p spaces in terms of the convexity of another function.

See also

• Katětov–Tong insertion theorem – On existence of a continuous function between semicontinuous upper and lower bounds
• Semicontinuous set-valued function

Bibliography

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