Question

Let \(A^{\prime}\) be the set of all cluster points of \(A \subseteq(S, \rho) .\) Let \(f: A \rightarrow\left(T, \rho^{\prime}\right)\) be uniformly continuous on \(A,\) and let \(\left(T, \rho^{\prime}\right)\) be complete. (i) Prove that \(\lim _{x \rightarrow p} f(x)\) exists at each \(p \in A^{\prime}\). (ii) Thus define \(f(p)=\lim _{x \rightarrow p} f(x)\) for each \(p \in A^{\prime}-A,\) and show that \(f\) so extended is uniformly continuous on the set \(\bar{A}=A \cup A^{\prime} .6\) (iii) Consider, in particular, the case \(A=(a, b) \subseteq E^{1},\) so that $$ \bar{A}=A^{\prime}=[a, b] $$

Short answer

f is uniformly continuous on \(\bar{A} = A \cup A'\).

Step-by-step solution

Understanding the Problem and Definitions

We begin by understanding that we have a set \(A\) within a metric space \((S, \rho)\) and we're considering its set of cluster points \(A'\). The function \(f\) is uniformly continuous from \(A\) to another complete metric space \((T, \rho')\). The task is to prove that for each cluster point \(p\), the limit \(\lim _{x \rightarrow p} f(x)\) exists and to extend \(f\) over \(\bar{A} = A \cup A'\) so that it remains uniformly continuous.

Proving Existence of Limit for Each Cluster Point

For each \(p \in A'\), it is a cluster point, meaning for any \(\epsilon > 0\), there are points \(x \in A\) such that \(0 < \rho(x, p) < \epsilon\). Since \(f\) is uniformly continuous on \(A\), for a given \(\epsilon' > 0\), there exists a \(\delta > 0\) such that for any \(x, y \in A\), if \(\rho(x, y) < \delta\), then \(\rho'(f(x), f(y)) < \epsilon'\). As \(x \to p\), \(\rho(x, p) < \delta\), thus \(f(x)\) approaches some limit \(L\) because \(T\) is complete. Hence \(\lim_{x \to p} f(x)\) exists.

Extending the Function

We define \(f(p) = \lim_{x \to p} f(x)\) for each cluster point \(p \in A' - A\). Now, we have a function \(f\) defined not just on \(A\) but also on its closure \(\bar{A} = A \cup A'\).

Proving Uniform Continuity on the Expanded Set

To show that \(f\), as extended, is uniformly continuous on \(\bar{A}\), consider any \(\epsilon' > 0\). Since \(f\) was originally uniformly continuous on \(A\), there was a \(\delta > 0\) ensuring that \(\rho'(f(x), f(y)) < \epsilon'\) for \(x, y \in A\) with \(\rho(x, y) < \delta\). For \(x \in A, p \in A'\), \(f(x) \to f(p)\) as \(x \to p\), maintaining \(\rho'(f(x), f(p)) < \epsilon'\) for sufficiently small \(\rho(x, p)\). Thus the extension remains continuous for all combinations within \(\bar{A}\).

Specializing the Case for Interval \((a, b)\)

Consider \(A = (a, b) \subseteq \mathbb{R}\). The cluster points \(A'\) include all limit points of \(A\), so \(A' = [a, b]\). The closure of \(A\), \(\bar{A} = [a, b]\), by definition, accommodates all limit points, including the endpoints. Through uniform continuity, along with analogy to the closure and horizon of real intervals, the extension maintains the Lipschitz condition across the expanded interval \([a, b]\).

Key concepts

Cluster Points

Cluster points are fundamental concepts in topology and analysis. When working with a subset \(A\) of a metric space \((S, \rho)\), a cluster point of \(A\) is a point \(p\) such that any neighborhood around \(p\) contains another point from \(A\). In simpler terms, you can get arbitrarily close to a cluster point with points from \(A\), but \(p\) does not have to be in \(A\) itself.
To better understand, let's imagine you have a collection of points scattered in a space. If you pick any point from this space, look at a small area around it, and find points from your collection no matter how tiny the area gets, then you have a cluster point. Cluster points help us understand the behavior and limits of functions on the boundaries of sets.
In mathematical analysis, proving the existence of limits at cluster points is crucial for extending functions and maintaining continuity.

Complete Metric Space

A metric space \((T, \rho')\) is called complete if every Cauchy sequence of points in \(T\) has a limit that is also in \(T\). Cauchy sequences are those that get arbitrarily close to each other as the sequence progresses. Imagine walking towards a wall, where every step halves the distance to the wall. In a complete space, you could potentially *reach* the wall by this process.
Completeness is essential in analysis because it ensures that limits exist within the space, which is particularly important when dealing with convergence of sequences or functions. This property allows us to handle limits, extensions of functions, and ensures that our spaces are "whole" without any missing points.
The concept of completeness was pivotal in our exercise as the range space \((T, \rho')\) needed to be complete for the limits of the function \(f\) to reliably exist at the cluster points.

Limit of a Function

The limit of a function at a point \(p\) is the value that the function \(f(x)\) approaches as \(x\) gets infinitely close to \(p\). If you can get \(f(x)\) as close as you want to some value \(L\) just by choosing \(x\) sufficiently close to \(p\), then \(L\) is the limit. In formal terms, for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(x\) is within \(\delta\) of \(p\), \(f(x)\) is within \(\epsilon\) of \(L\).
In the exercise, we determined that \(\lim_{x \to p} f(x)\) exists for each cluster point \(p\). This is vital because it allows the extension of \(f\) beyond its original domain \(A\), onto the closure \(\overline{A}\), while maintaining consistency in value and continuity.
Limits ensure that even at the boundary points of your domain where functions are extended, there is a seamless value that the function approaches, maintaining the function's integrity and behavior.

Extension of Functions

Extending a function means broadening its domain, while making sure it stays continuous and well-behaved. In the exercise, we extended the uniformly continuous function \(f\) from its original domain \(A\) to include its cluster points \(A'\), resulting in a new domain \(\bar{A} = A \cup A'\).
This extension wasn't arbitrary. For the extension to remain uniformly continuous, the limit at each cluster point had to exist and be well-defined, ensuring that the same \(\epsilon-\delta\) conditions that held for uniform continuity before still hold now. The key was proving that the limit \(\lim_{x \to p} f(x)\) exists at every cluster point \(p\). In essence, extending functions allows us to describe their behavior consistently and uniformly over larger sets.
Overall, extending functions involves careful calculations and proofs but helps in analyzing functions beyond their initial domain, maintaining characteristics like continuity and uniformity across new sets.