View source for Semicontinuous function

{{MSC|54C05|54A05}}
[[Category:Topology]]
{{TEX|done}}

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|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 (cp. with Section 6.2 of Chapter IV in {{Cite|Bo}}).

'''Definition 3'''
Let $X$ be a [[Topological space|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 (cp. with Proposition 1 of {{Cite|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 {{Cite|Ba1}} for functions of one real variable (cp. with [[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 a lower and upper 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|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.

(Cp. with  Proposition 2 and Theorem 4 in Chapter IV of {{Cite|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 (cp. with Lemma 2.2.9 of {{Cite|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 (cp. with [[Baire theorem]]). Thus, under these assumptions, $f$ belongs to the first [[Baire classes|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 (cp. with Theorem 7.6 of Chapter 3 in {{Cite|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 increasing and $\{v_n (x)\}$ is bounded above and monotone decreasing 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==
{|
|valign="top"|{{Ref|Ba2}}||  R. Baire,   "Leçons sur les  fonctions discontinues, professées au  collège de France" ,  Gauthier-Villars  (1905) {{ZBL|36.0438.01}}
|-
|valign="top"|{{Ref|He}}|| L. Helms, "Potential theory". Universitext. Springer-Verlag London, Ltd., London,  2009. {{MR|2526019}} {{ZBL|1180.31001}} 
|-
|valign="top"|{{Ref|HS}}||  E. Hewitt,   K.R. Stromberg,   "Real and  abstract analysis" , Springer  (1965) {{MR|0188387}} {{ZBL|0137.03202}}
|-
|valign="top"|{{Ref|Ke}}||  J.L. Kelley,   "General topology" , v. Nostrand  (1955) {{MR|0070144}} {{ZBL|0066.1660}} 
|-
|valign="top"|{{Ref|Na}}|| I.P. Natanson,    "Theory of functions of a real variable" , '''1–2''' , F. Ungar   (1955–1961) {{MR|0067952}} {{ZBL|0064.29102}} 
|-
|valign="top"|{{Ref|Ru}}||  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1964) {{MR|038502}}  {{ZBL|0346.2600}}
|-
|valign="top"|{{Ref|Sa}}|| S. Saks,    "Theory of the integral" , Hafner  (1937) {{MR|0167578}} {{ZBL|63.0183.05}}
|-
|}