Problem 6 - Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Question

1-28. If \(A \subset \mathbf{R}^{n}\) is not closed, show that there is a continuous function \(f: A \rightarrow \mathbf{R}\) which is unbounded. Hint: If \(x \in \mathbf{R}^{n}-A\) but \(x \notin\) interior \(\left(\mathbf{R}^{n}-A\right)\), let \(f(y)=1 /|y-x|\).

Step-by-Step Solution

Answer

The function \( f(y) = \frac{1}{|y - x|} \) is continuous and unbounded on \( A \).

Step 1: Identify the Set and Its Properties

Consider a set \(A \subset \mathbf{R}^{n}\) that is not closed. This means that there exists at least one point \(x \in \mathbf{R}^{n}\) that is a limit point of \(A\) but \(x otin A\).

Step 2: Define the Function

Use the hint provided. Define a function \(f: A \rightarrow \mathbf{R}\) by \(f(y) = \frac{1}{|y - x|}\), where \(x \) is a point in \(\mathbf{R}^{n} - A\) but not in the interior of \( \mathbf{R}^{n} - A\). This means such \( x \) is a boundary point of \( A \).

Step 3: Verify Continuity of the Function

The function \( f(y) = \frac{1}{|y - x|} \) is continuous on \(A\) because it is the composition of continuous functions (the distance function and the reciprocal function where the argument is non-zero).

Step 4: Show Unboundedness

To show that the function \( f(y) \) is unbounded, examine its behavior as \( y \) approaches \( x \). Since \( x \) is a limit point of \( A \), there are points \( y \in A \) arbitrarily close to \( x \). As \( y \) gets closer to \( x \), \( |y - x| \) approaches 0, and thus \( f(y) = \frac{1}{|y - x|} \) becomes arbitrarily large. Hence, \( f(y) \) is unbounded.

Key Concepts

Limit Points

In topology, a limit point (or accumulation point) of a set A is a point x such that any neighborhood of x contains at least one point from A different from x itself. In simpler terms, it's a point that can be approached by other points within the set. For example, in the context of the exercise, if set A is not closed, it means that there is at least one limit point of A, but this point is not included in A. This is an essential aspect of the problem that drives the solution forward.

Continuous Functions

A function is continuous if small changes in the input result in small changes in the output. Formally, a function f is continuous at a point y if the limit of f as y approaches a point equals the function value at that point. This property is crucial for defining the function in the exercise. We define the function as f(y) = \( \frac{1}{|y - x|} \). Both the distance function and the reciprocal function are continuous. Therefore, their composition remains continuous.

Unboundedness

A function is unbounded if its values can grow arbitrarily large. For the function f(y) = \( \frac{1}{|y - x|} \), as y approaches the point x, the distance |y - x| gets very small. Since we're dividing 1 by an increasingly smaller number, the function f(y) can become arbitrarily large. This characteristic shows that f is unbounded, fulfilling the requirement of the original problem.

Distance Function

The distance function calculates the distance between two points in a given space. In our case, the function is |y - x|, which finds the distance between the points y and x. The absolute value is used because distance is always a non-negative value. The role of the distance function in this exercise is to ensure our function correctly evaluates the proximity between points, which is crucial for determining how the function behaves as y approaches x.

Reciprocal Function

The reciprocal function takes a number and returns its reciprocal, which is 1 divided by that number. In the context of our function, this reciprocal aspect is what allows f(y) to become unbounded. As the distance |y - x| becomes very small, the reciprocal—that is, 1 divided by this small distance—becomes very large. This concept perfectly ties in with demonstrating the unbounded nature of the function f(y) = \( \frac{1}{|y - x|} \).