Chapter 3 - Topological Spaces
3.12 Real Functions on Metric and Topological Spaces
Results
Functions and Functionals
Let $T$ be a topological space, in particular a metric space. A real function is a mapping
$f: T\rightarrow\R$. If $T$ is a function space, a space of functions, then a real function
on $T$ is called a
functional. Examples:
\begin{align}
F_1(x) &= \sup_{t\in[0,1]}x(t) \\
F_2(x) &= \int_0^1|x'(t)|dt
\end{align}
Definition 1
A real function $f(x)$ defined on a metric space $R$ is said to be
uniformly continuous on $R$,
if given any $\eps > 0$, there is a $\delta > 0$ such that $\rho(x_1, x_2) < \delta$ implies
$|f(x_1) - f(x_2)| < \eps$ for all $x_1,x_2\in R$.
Theorem 1
A real function $f$ continuous on a compact metric space $R$ is uniformly continuous on $R$.
Proof.
Suppose $f$ is continuous but not uniformly continuous on $R$. Then for some positive $\eps$ and every $n$
there are points $x_n$ and $x_n'$ in $R$ such that
$$
\rho(x_n, x_n') < \frac{1}{n},
\tag{1}
$$
but
$$
|f(x_n) - f(x_n')| \geq \eps.
\tag{2}
$$
Since $R$ is compact, the sequence $\{x_n\}$ has a subsequence $\{x_{n_k}\}$ converging to a point $x\in R$.
Hence $\{x'_{n_k}\}$ also converges to $x$, because of (1). But then at least one of the inequalities
$$
|f(x) - f(x_{n_k})|\geq\frac{\eps}{2},\quad
|f(x) - f(x'_{n_k})|\geq\frac{\eps}{2}
$$
must hold for arbitrary $k$, because of (2). This contradicts the assumed continuity of $f$ at $x$.
■
Theorem 2
A real function $f$ continuous on a compact topological space $T$ is bounded on $T$. Moreover, $f$ achieves
its least upper bound and greatest lower bound on $T$.
Definition 2
A real function $f$ defined on a topological space $T$ is said to be
upper semicontinuous at a point
$x_0\in T$ if, given $\eps > 0$, there exists a neighborhood of $x_0$ in which $f(x) < f(x_0) + \eps$.
Similarly, $f$ is said to be
lower semicontinuous at $x_0$, if, given $\eps > 0$, there exists a
neighborhood of $x_0$ in which $f(x) > f(x_0) - \eps$.
Theorem 2'
A finite lower semicontinuous function $f$ defined on a compact topological space $T$ is bounded from below.
Theorem 2''
A finite lower semicontinuous function $f$ defined on a compact topological space $T$ achieves its greatest
lower bound on $T$.
Definition 3
Given a real function $f$ defined on a metric space $R$, the (finite or infinite) quantity
$$
\overline{f}(x_0) = \lim_{\eps\rightarrow 0}\left\{\sup_{x\in S(x_0, \eps)} f(x)\right\}
$$
is called the
upper limit of $f$ at $x_0$, while the (finite or infinite) quantity
$$
\underline{f}(x_0) = \lim_{\eps\rightarrow 0}\left\{\inf_{x\in S(x_0, \eps)} f(x)\right\}
$$
is called the
lower limit of $f$ at $x_0$. The difference
$$
\omega f(x_0) = \overline{f}(x_0) - \underline{f}(x_0),
$$
provided that it exists (one of the numbers are finite), is called the
oscillation of $f$ at $x_0$.
Skipping these exercises for now.