Problem 7 - Advanced Calculus

Question

Let \(D\) be a bounded set and let \(f\) be uniformly continuous on \(D \subset \mathbf{R}^{n} .\) Prove that \(f\) is bounded on \(D\).

Step-by-Step Solution

Answer

The uniformly continuous function \(f\) on the bounded set \(D\) is also bounded as the rate of change of the function remains bounded across its domain and the set \(D\) can be covered using a finite number of \(\delta\)-balls. The function \(f\) can thus be said to be bounded by \(M\), with \(M = \max_{1 \leq i \leq n} |f(x_i)| + 1\).

Step 1: Understand the Definitions

Uniform continuity refers to the idea that the rate of change of the function remains bounded across its domain. For a function \(f\) to be uniformly continuous on a set \(D\), it means that for every \(\epsilon > 0\) there exists a \(\delta > 0\) such that for all \(x, y\) in \(D\), if \(\|x - y\| < \delta\), then \(\|f(x) - f(y)\| < \epsilon\). A set \(D\) is bounded if there exists a real number \(M\) such that \(|x| ≤ M\) for every \(x\) in \(D\). Similarly, a function \(f\) is bounded if there exist real numbers \(m\) and \(M\) such that \(m \leq f(x) \leq M\) for all natural numbers \(x\) in \(D\).

Step 2: Using Uniform Continuity

Choose an \(\epsilon = 1\) (this can be any positive number), then by the definition of uniform continuity there exists a \(\delta > 0\) such that if \(\|x - y\| < \delta\), \(\|f(x) - f(y)\| < 1\). This is the starting point of our proof.

Step 3: Constructing a Finite Cover

Since \(D\) is bounded, there exists an open ball that contains \(D\). Pick a point \(x_0\) in \(D\). We can construct a finite number of \(\delta\)-balls centered at points \(x_0, x_1, ..., x_n\) in \(D\) whose union covers \(D\) completely. This indicates that the set \(D\) can be covered using a finite number of these \(\delta\)-balls.

Step 4: Bounding the Function f

Find \(M = \max_{1 \leq i \leq n} |f(x_i)| + 1\). For any \(x\) in \(D\), \(x\) is covered by some \(\delta\)-ball \(B(x_i, \delta)\). Thus, by using the uniform continuity of \(f\), we have: \(|f(x)| = |f(x) - f(x_i) + f(x_i)| \leq |f(x) - f(x_i)| + |f(x_i)| < 1 + |f(x_i)| \leq M\). Therefore, \(f\) is bounded on \(D\).

Key Concepts

Uniform Continuity

Understanding uniform continuity is essential, as it has significant implications for the behavior of functions across their entire domain. A function is said to be uniformly continuous on a set if small changes in the input lead to correspondingly small changes in the output, regardless of where you are in the set.

In more technical terms, for every tiny positive number we choose (\textbf{epsilon}), we can find another tiny positive number (\textbf{delta}) such that if any two points are within delta distance of each other, then their outputs will be within epsilon distance. We express this mathematically as \(\forall \epsilon > 0, \exists \delta > 0:\ \|x - y\| < \delta \Rightarrow \|f(x) - f(y)\| < \epsilon\). This concept assures us that the function behaves 'nicely' and predictably in its entirety on domain \(D\).

Bounded Set

When we refer to a 'bounded set' in the real number space, we’re talking about a space where all of its elements stay within certain limits.

Imagine holding a rope—the rope has a beginning and an end. Similarly, a bounded set has 'bounds' and all points within it will not exceed those bounds. Formally, there's a number \(M\) that's bigger than every single point in the set. Mathematically, if \(D\) is bounded, then \(\exists M \in \mathbf{R}: \forall x \in D, \|x\| \leq M\). Knowing that a set is bounded provides structure and limitations which can be very useful in analysis.

Real Number Space

The real number space, denoted as \(\mathbf{R}^{n}\), is a fascinating landscape where each point is an 'n-tuple' of real numbers. This means we're working with not just a line (\textbf{1D}), but possibly a plane (\textbf{2D}), space (\textbf{3D}), and so on.

Think of the real number space as a vast ocean where every point pinpoints a location using a set of coordinates and every coordinate is a real number. For example, the position of a particle in space can be described with three coordinates (\textbf{3D}): \( (x, y, z) \) within this space.

Finite Cover

A finite cover is a collection of 'patches' that entirely blanket a set, and importantly, there's a set number of these patches—hence 'finite'.

In our context, think of it as covering a floor with a finite number of tiles; if you can cover every inch of the floor with tiles and count them, that's a finite cover. Specifically, for a set \(D\) and a given distance \(\delta\), we can cover \(D\) with a limited number of open balls (our 'tiles') of radius \(\delta\), centered on points within \(D\). Each point in \(D\) lies inside at least one of these \(\delta\)-balls.