# Semicontinuous function

2010 Mathematics Subject Classification: Primary: 54C05 Secondary: 54A05 [MSN][ZBL]

A concept in analysis and topology, related to the real-valued functions.

## Definition

### Functions of one real variable

The concept of semicontinuous function was first introduced for functions of one variable, using upper and lower limits.

Definition 1 Consider a function $f:\mathbb R\to\mathbb R$ and a point $x_0\in\mathbb R$. The functiom $f$ is said to be upper (resp. lower) semicontinuous at the point $x_0$ if $f (x_0) \geq \limsup_{x\to x_0}\; f(x) \qquad \left(\mbox{resp. }\quad f(x_0)\leq \liminf_{x\to x_0}\; f(x)\right)\, .$

The definition can be easily extended to functions defined on subdomains of $\mathbb R$ and taking values in the extended real line $[-\infty, \infty]$. If a function is upper (resp. lower) semicontinuous at every point of its domain of definition, then it is simply called an upper (resp. lower) semicontinuous function.

### Extensions

The definition can be easily extended to functions $f:X\to [-\infty, \infty]$ where $(X,d)$ is an arbitrary metric space, using again upper and lower limits. Observe further that the following holds:

Proposition 2 Let $(X, d)$ be a metric space, $f: X\to [-\infty, \infty]$ and $x_0\in X$. Then $f$ is upper (resp. lower) semicontinuous at $x_0$ if and only if, either $f(x_0) =\infty$ (resp. $f(x_0)=-\infty$), or $\{f< a\} \quad (\mbox{resp.}\;\; \{f>a\}) \quad \mbox{ is a neighborhood of } x_0 \mbox{ for all } a> f(x_0) \mbox{ (resp. } a<f(x_0))\, .$

This motivates the following general definition (cf. Section 6.2 of Chapter IV in [Bo]).

Definition 3 Let $X$ be a topological space, $f:X\to [-\infty, \infty]$ and $x_0\in X$. $f$ is upper (resp. lower) semicontinuous at $x_0$ if, either $f(x_0) =\infty$ (resp. $f(x_0)=-\infty$), or $\{f< a\} \quad (\mbox{resp.}\;\; \{f>a\}) \quad \mbox{ is a neighborhood of } x_0 \mbox{ for all } a> f(x_0) \mbox{ (resp. } a<f(x_0)) \, .$

As an obvious consequence of the above definition, we have (cf. Proposition 1 of Section 6.2 in Chapter IV of [Bo])

Theorem 4 Let $X$ be a topological space. A function $f:X\to [-\infty, \infty]$ is upper (resp. lower) semicontinuous if and only if $\{f\geq a\}$ (resp. $\{f\leq a\}$) is closed for every $a$.

The latter theorem was first proved by R. Baire in [Ba] for functions of one real variable (cf. Baire theorem).

## Properties

### Relations to continuous functions

A function $f$ is continuous at $x_0$ if and only if it is both upper and lower semicontinuous at that point. Conversely we have the following

Proposition 5 If $u$ and $v$ are an upper and a lower semicontinuous function on a complete metric space $(X,d)$ such that $-\infty<u(x)\leq v(x)<\infty$ for every $x\in X$, then there is a continuous function $f:X\to\infty$ such that $u\leq f\leq v$.

### Existence of extrema

Moreover, we have the following important

Theorem 6 If its domain of definition is a compact topological space an upper (resp. lower) semicontinuous function achieves a maximum (resp. a minimum).

### Upper and lower semicontinuous envelopes

The space of upper (resp. lower) semicontinuous functions on a topological space $X$ is a lattice. More precisely

Theorem 7 Let $\mathcal{F}$ be an arbitrary family of upper (resp. lower) semicontinuous functions on a given topological space $X$. Then the function $F(x):= \inf_{f\in\mathcal{F}}\; f(x) \qquad \left(\mbox{resp.}\; \sup_{f\in\mathcal{F}}\; f(x)\right)$ is upper (resp. lower) semicontinuous.

(Cf. Proposition 2 and Theorem 4 of Section 6.2 in Chapter IV of [Bo]). Therefore we can define

Definition 8 Let $X$ be a topological space. The upper (resp. lower) semicontinuous envelope of a function $f: X\to [-\infty,\infty]$ is the smallest (resp. largest) upper (resp. lower) semicontinuous function $g$ such that $g\geq f$ (resp. $g\leq f$).

### Dini-Cartan lemma

A very useful fact on semi-continuous functions is the Dini–Cartan lemma (cf. Lemma 2.2.9 of [He]).

Theorem 9 Let $X$ be a compact topological space and $\{f_i\}_{i\in I}$ a family of upper (resp. lower) semicontinuous function on $X$ such that, for every finite $J\subset I$ there is $i_0\in I$ with $\inf_{j\in J} f_j \geq f_{i_0}$. (resp. $\sup_{j\in J} \leq f_{i_0}$). If $g$ is a lower (resp. upper) semicontinuous function such that $g> \inf_{i\in I}\, f_i$ (resp. $g< \sup_{i\in I}\, f_i$), then there is a $j\in I$ such that $g> f_j$ (resp. $g< f_j$). In particular $\sup_{x\in E}\; \inf_{i\in I}\; f_i (x) = \inf_{i\in I}\;\sup_{x\in E}\; f_i (x)$ (resp. $\inf_{x\in E}\; \sup_{i\in I}\; f_i (x) = \sup_{i\in I}\;\inf_{x\in E}\; f_i (x) \Big)\, .$

## Relation to Baire classes

If $X$ is a completely regular space and $f$ an upper (resp. lower) semicontinuous function which does not take the value $\infty$ (resp. $-\infty$), then $f$ is the pointwise limit of a monotone nonincreasing (resp. nondecreasing) sequence of continuous functions (cf. Baire theorem). Thus, under these assumptions, $f$ belongs to the first Baire class. However not every function in the first Baire class is semicontinuous: for instance the following function $f:\mathbb R\to\mathbb R$ is neither upper semicontinuous nor lower semicontinuous: $f(x) = \left\{ \begin{array}{ll} 1 \qquad &\mbox{if } x>0\\ 0 \qquad &\mbox{if } x=0\\ -1 \qquad &\mbox{if } x<0\, . \end{array}\right.$ Nonetheless, being $f$ the pointwise limit of the sequence of continuous functions $f_n(x) =\arctan (nx)$, it belongs to the first Baire class.

## Vitali-Caratheodory theorem

A theorem relating semicontinuous functions to measurable ones (cf. Theorem 7.6 of Chapter 3 in [Sa]).

Theorem 10

If $\mu$ is a non-negative regular Borel measure on $\mathbb R^n$, then for any $\mu$-measurable function $f$ there exist two sequences of functions $\{u_n\}$ and $\{v_n\}$ satisfying the following conditions:

1) $u_n$ is lower semi-continuous and $v_n$ is upper semi-continuous;

2) $\{u_n (x)\}$ is bounded below and monotone decreasing and $\{v_n (x)\}$ is bounded above and monotone increasing for every $x$;

3) $u_n (x)\geq f(x)\geq v_n (x)$ for all $x$;

4) For $\mu$-a.e. $x$ we have $u_n (x)\downarrow f(x)$ and $v_n (x)\uparrow f(x)$;

5) If in addition $f\in L^1 (\mu)$, then $u_n, v_n\in L^1 (\mu)$ and $\lim_{n\to\infty} \int u_n\, d\mu = \lim_{n\to\infty} \int v_n\, d\mu = \int f\, d\mu\, .$

How to Cite This Entry:
Semicontinuous function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semicontinuous_function&oldid=40145
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article