
Chapter 3 - Topological Spaces

3.12 Real Functions on Metric and Topological Spaces

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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.