Topological Spaces: Real Functions on Metric and Topological Spaces
Definition Let \(T\) be a topological space (in particular a metric space). A real function on \(T\) is a mapping of \(T\) into the space \(R^1\) (the real line).
Defintion A space whose elements are functions is called a function space. A real function on a function space is called functional.
Definition A real function \(f(x)\) defined on a metric space \(R\) is said to be uniformly continuous on \(R\) if, given any \(\epsilon > 0\), there is a \(\delta > 0\) such that \(\rho(x_1, x_2) < \delta\) implies \(|f(x_1) - f(x_2)| < \epsilon\) 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 \(\epsilon\) and every \(n\) there are points \(x_n\) and \(x’_{n}\) in \(R\) such that \[\rho(x_n, x’_n) < \frac{1}{n} \quad (1)\] but \[|f(x_n) - f(x’_n)| \geq \epsilon. \quad (2)\]
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 \[|f(x) - f(x_{n_k})| \geq \frac{\epsilon}{2}, \quad |f(x) - f(x’_{n_k})| \geq \frac{\epsilon}{2}\] must hold for arbitrary \(k\), because of \((2)\). This contradicts the assumed continuity of \(f\) at \(x\).
Definition A real function (or functional) \(f\) is said to be bounded on a set \(E\) if \(f(E)\) is contained in some interval \([-C, C]\).
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 \(T\).
Proof
A continuous real function on \(T\) is a continuous mapping of \(T\) into the real line \(R^1\). The image of \(T\) in \(R^1\) is compact, by Theorem 5, page 94. But every compact subset of \(R^1\) is bounded and closed (see page 101). Hence \(f\) is bounded on \(T\). Moreover, \(f\) not only has a least upper bound and greatest lower bound on \(T\), but actually achieves these bounds at points of \(T\).
Definition 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 any \(\epsilon > 0\), there exists a neighbourhood of \(x_0\) in which \(f(x) < f(x_0) + \epsilon\). Similarly, \(f\) is said to be lower semicontinuous at \(x_0\) if, given any \(\epsilon > 0\), there exists a neighbourhood of \(x_0\) in which \(f(x) > f(x_0) - \epsilon\).
Theorem 2'
A finite lower semicontinuous function \(f\) defined on a compact topological space \(T\) is bounded from below. (Note that finite means that the \(f(x)\) is finite for any point \(x \in T\).)
Proof
Suppose to the contrary that \(\inf f(x) = - \infty\). Then there exists a sequence \(\{x_n\}\) such that \(f(x_n) < -n\). Since \(T\) is compact, the infinite set \(E = \{x_1, \ldots, x_n, \ldots\}\) has at least one limit point \(x_0\). Since \(f\) is finite and lower semicontinuous at \(x_0\), there is a neighbourhood \(U\) of \(x_0\) in which \(f(x) > f(x_0) - 1\). But then \(U\) can only contain finitely many points of \(E\), so that \(x_0\) cannot be a limit point of \(E\).
Theorem 2''
A finite lower semicontinuous function \(f\) defined on a compact topological space \(T\) achieves its greatest lower bound on \(T\).
Proof
By Theorem 2', \(\inf f(x)\) is finite. Clearly, there exists a sequence \(\{x_n\}\) such that \[f(x_n) \leq \inf f(x) + \frac{1}{n}.\] By the compactness of \(T\), the set \(E = \{x_1, \ldots, x_n, \ldots\}\) has at least one limit point \(x_0\). If \(f(x_0) > \inf f\), then, by the semicontinuity of \(f\) at \(x_0\), there is a neighbourhood \(U\) of the point \(x_0\) and a \(\delta> 0\) such that \(f(x) > \inf f + \delta\) for all \(x \in U\). But then \(U\) cannot contain an infinite subset of \(E\), i.e., \(x_0\) cannot be a limit point of \(x_0\). It follows that \(f(x_0) = \inf f\).
Definition Given a real function \(f\) defined on a metric space \(R\), the (finite or infinite) quantity \[\overline{f}(x_0) = \lim_{\epsilon\rightarrow 0}\left\{\sup_{x\in S(x_0, \epsilon)} f(x)\right\}\] is called the upper limit of \(f\) at \(x_0\), while the (finite or infinite) quantity \[\underline{f}(x_0) = \lim_{\epsilon\rightarrow 0}\left\{\inf_{x\in S(x_0, \epsilon)} f(x)\right\}\] 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 it exists (i.e. at least one of them is finite, is called the oscillation of \(f\) at \(x_0\).
Definition Two continuous functions \(P = f(t')\), \(P = g(t'')\), defined on intervals \(a' \leq t' \leq b\), \(a'' \leq t'' \leq b''\) and taking values in a metric space \(R\), are said to be equivalent if there exist two continuous nondecreasing functions \(t’ = \phi(t)\), \(t'' = \psi(t)\), defined on the same interval \(a \leq t \leq b\), such that \[\phi(a) = a', \quad \phi(b) = b'\] \[\psi(a) = a'', \quad \psi(b) = b''\] and \(f(\phi(t)) = g(\psi(t))\) for all \(t \in [a, b]\). As this relation of equivalence is reflexive, symmetric and transitive, the set of all continuous functions of the given type can be partitioned into classes of equivalent functions, and each such class is said to define a (continuous) curve in the space \(R\).
Note: One way to think about whether two continuous functions are equivalent or not, is whether they are stretched (or even reflected) along the x-axis. No such stretching may occur in the y-axis.
