Semicontinuous function
2020 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 \[ \lim_{n\to\infty} \int u_n\, d\mu = \lim_{n\to\infty} \int v_n\, d\mu = \int f\, d\mu\, . \]
References
[Ba] | R. Baire, "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars (1905) Zbl 36.0438.01 |
[Bo] | N. Bourbaki, "General topology: Chapters 5-10", Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998. MR1726872 Zbl 0894.54002 |
[He] | L. Helms, "Potential theory". Universitext. Springer-Verlag London, Ltd., London, 2009. MR2526019 Zbl 1180.31001 |
[HS] | E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202 |
[Ke] | J.L. Kelley, "General topology" , v. Nostrand (1955) MR0070144 Zbl 0066.1660 |
[Na] | I.P. Natanson, "Theory of functions of a real variable" , 1–2 , F. Ungar (1955–1961) MR0067952 Zbl 0064.29102 |
[Ru] | W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1964) MR038502 Zbl 0346.2600 |
[Sa] | S. Saks, "Theory of the integral" , Hafner (1937) MR0167578 Zbl 63.0183.05 |
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