Question

Let \(E\) be a dense subset of a metric space \(X\), and let \(f\) be a uniformly continuous real function defined on \(E\). Prove that \(f\) has a continuous extension from \(E\) to \(X\) (see Exercise 5 for terminology). (Uniqueness follows from Exercise 4.) Hint: For each \(p \in X\) and each positive integer \(n\), let \(V_{n}(p)\) be the set of all \(q \in E\) with \(d(p, q)<1 / n .\) Use Exercise 9 to show that the intersection of the closures of the sets \(f\left(V_{1}(p)\right), f\left(V_{2}(p)\right), \ldots\), consists of a single point, say \(g(p)\), of \(R^{1}\). Prove that the function \(g\) so defined on \(X\) is the desired extension of \(f\). Could the range space \(R^{1}\) be replaced by \(R^{k} ?\) By any compact metric space? By any complete metric space? By any metric space?

Short answer

Question: Prove that there exists a continuous extension of a uniformly continuous real function f defined on a dense subset E of a metric space X. Consider possible generalizations and the uniqueness of the extension. Answer: A continuous extension of f from E to X exists, defined as g(p) at each p in X by taking the single point intersection of the closures of sets f(V_n(p)) for positive integer values of n. This extension, g, is continuous on X and is equal to f when restricted to E. The method can be generalized for the cases where the range space is R^k or compact metric spaces. The uniqueness of the extension follows from the fact that a continuous function has a unique continuous extension from a dense subset to a complete metric space, as stated in Exercise 4.

Step-by-step solution

Define the function g for each p in X

Given any positive integer n, we denote \(V_n(p)\) as the set of all \(q \in E\) with \(d(p, q) < \frac{1}{n}.\) For each \(p \in X\), using Exercise 9 we show that the intersection of the closures of the sets \(f\left(V_{1}(p)\right), f\left(V_{2}(p)\right), \ldots\) consists of a single point, say \(g(p)\), of \(R^1\). Now, g is defined for each p in X.

Prove that g is a continuous extension of f

To prove that g is a continuous extension of f, we need to show two things - first, that g is continuous on X, and second, that g restricted to E is equal to f. We can do this by showing that given any \(\epsilon > 0,\) there exists a \(\delta > 0\) such that \(|g(p) - g(q)| < \epsilon\) for all \(p, q \in X\) with \(d(p, q) < \delta\). This would imply that g is continuous on X and that g restricted to E is equal to f, satisfying all conditions for a continuous extension.

Consider possible generalizations

By following the same method, we can extend the result to the case when the range space is \(R^k\) instead of \(R^1\). For compact metric spaces as the range space, we can also extend the result by using properties of compactness. However, for general and complete metric spaces as the range space, the method in this exercise may not work directly, so further investigation is needed.

Refer to Exercise 4 for uniqueness

It is mentioned in the exercise that uniqueness follows from Exercise 4, which states that a continuous function has a unique continuous extension from a dense subset to the complete metric space. Hence, the extension we found in this exercise is unique.

Key concepts

Metric Space

A metric space is a way to describe spaces in mathematics where we can discuss the distance between points. It is comprised of a set along with a metric, which is a function that defines a distance between any two points in the set. This metric has to satisfy the following properties:

• Non-negativity: The distance between two points is always greater than or equal to zero.
• Identity of indiscernibles: The distance between two points is zero if and only if they are the same point.
• Symmetry: The distance from point A to point B is the same as from point B to point A.
• Triangle inequality: The distance between two points is always less than or equal to the sum of the distances of the parts of any path that includes those two points.

Understanding metric spaces is fundamental as they provide a framework for discussing the concepts of convergence, continuity, and other important topological properties. For example, in our exercise, the metric space forms the backdrop against which continuous functions and their properties are examined.

Dense Subset

A dense subset in a metric space is a subset that is, in a sense, tightly packed within the metric space. Formally, a subset \( E \) of a metric space \( X \) is dense in \( X \) if every point in \( X \) is either in \( E \) or is a limit point of \( E \). This means that for every point \( p \) in \( X \), no matter how small a neighborhood you look at around \( p \), you will always find at least one point from \( E \).

Dense subsets are crucial in analysis because they allow extensions of properties from the subset to the entire space. In our exercise, the dense subset \( E \) allows the uniformly continuous function \( f \) to be extended beyond the subset \( E \) to the entire metric space \( X \) seamlessly. This ensures the function maintains continuity as it reaches into more of the space.

Continuous Extension

In mathematics, extending a function means broadening its domain while preserving certain properties. A continuous extension of a function involves going from a smaller domain to a larger domain where the function remains continuous. In our context, this concept is about extending a uniformly continuous function from a dense subset \( E \) of a metric space \( X \) to the whole space \( X \).

To ensure the extension is continuous, the function must satisfy the following:

• On the subset \( E \), the extended function should coincide with the original function.
• As the extended function is defined on \( X \), it should retain the property of continuity: small changes in the input result in small changes in the output.

In the exercise provided, the method of defining \( g(p) \), a new function on \( X \), cleverly uses the fact that \( f \) is uniformly continuous and builds the extension through careful selection and intersection of image closures.

Compact Metric Space

A compact metric space is a specific type of metric space with interesting and useful properties. Informally, a space is compact if it is both limited in extent and contains all of its limit points. More formally, a metric space \( X \) is compact if every open cover of \( X \) has a finite subcover.

This property of compactness leads to several convenient results, such as every sequence in \( X \) having a convergent subsequence. When it comes to functions, a continuous function from a compact space into a metric space is often much easier to handle. For instance, such functions are not only continuous but also uniformly continuous and have manageable ranges.

In the exercise in question, exploring the possibility of having a compact metric space as the range shows that the extension properties and behaviors of functions can sometimes be generalized, provided the range maintains compactness. This property allows functions to be effectively extended while retaining desirable characteristics.