Question

We say that \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) is uniformly continuous on \(S\) if each of its components is uniformly continuous on \(S\). Prove: If \(\mathbf{F}\) is uniformly continuous on \(S,\) then for each \(\epsilon>0\) there is a \(\delta>0\) such that $$ |\mathbf{F}(\mathbf{X})-\mathbf{F}(\mathbf{Y})|<\epsilon \quad \text { if } \quad|\mathbf{X}-\mathbf{Y}|<\delta \quad \text { and } \quad \mathbf{X}, \mathbf{Y} \in S. $$

Short answer

To prove that a vector function \(\mathbf{F}:\mathbb{R}^n \rightarrow \mathbb{R}^m\) is uniformly continuous on a set \(S\), we first express \(\mathbf{F}\) as a set of uniformly continuous components \(f_i\). Then, for a given \(\epsilon > 0\), we find a suitable \(\delta\) value defined as the minimum of the \(\delta_i\) values associated with each component. By using the triangle inequality on each component and on the vector function as a whole, we show that for any \(\mathbf{X}, \mathbf{Y} \in S\), if \(|\mathbf{X} - \mathbf{Y}| < \delta\), then \(|\mathbf{F}(\mathbf{X}) - \mathbf{F}(\mathbf{Y})| < \epsilon\), proving that \(\mathbf{F}\) is uniformly continuous on \(S\).

Step-by-step solution

Write the definition of uniformly continuous components

Let \(\mathbf{F}=(f_1, f_2, \dots, f_m)\), where each \(f_i:\mathbb{R}^n \rightarrow \mathbb{R}\) is uniformly continuous on \(S\). That is, for each \(\epsilon > 0\), there exists a \(\delta_i > 0\) such that for any \(\mathbf{X}, \mathbf{Y} \in S\), if \(|\mathbf{X} - \mathbf{Y}| < \delta_i\), then \(|f_i(\mathbf{X}) - f_i(\mathbf{Y})| < \frac{\epsilon}{\sqrt{m}}\).

Define the desired delta value

Let \(\delta = \min(\delta_1, \delta_2, \dots, \delta_m)\). Note that \(\delta > 0\) since each \(\delta_i > 0\).

Use the triangle inequality on each component of \(\mathbf{F}\)

Now, let \(\mathbf{X}, \mathbf{Y} \in S\) such that \(|\mathbf{X} - \mathbf{Y}| < \delta\). Since \(\delta \leq \delta_i\) for each \(i\), it follows that \(|\mathbf{X} - \mathbf{Y}| < \delta_i\). Hence, for each \(i\), we have \(|f_i(\mathbf{X}) - f_i(\mathbf{Y})| < \frac{\epsilon}{\sqrt{m}}\).

Apply the triangle inequality to the whole vector function \(\mathbf{F}\)

Now, consider the function \(|\mathbf{F}(\mathbf{X}) - \mathbf{F}(\mathbf{Y})|^2 = |(f_1(\mathbf{X}) - f_1(\mathbf{Y}), \dots, f_m(\mathbf{X}) - f_m(\mathbf{Y}))|^2\). By the triangle inequality, we have: $$|\mathbf{F}(\mathbf{X}) - \mathbf{F}(\mathbf{Y})|^2 \leq \sum_{i=1}^m |f_i(\mathbf{X}) - f_i(\mathbf{Y})|^2 < \sum_{i=1}^m \frac{\epsilon^2}{m} = \epsilon^2$$. Since both sides of the inequality are non-negative, we can take the square root of both sides: $$|\mathbf{F}(\mathbf{X}) - \mathbf{F}(\mathbf{Y})| < \epsilon$$. This proves that \(\mathbf{F}\) is uniformly continuous on \(S\) according to the given definition.

Key concepts

Real Analysis

Real Analysis is a branch of mathematical analysis dealing with real numbers and the analytical properties of real-valued sequences and functions. It involves rigorous examination of the concepts of convergence, continuity, and differentiation among others.

In the context of Real Analysis, continuity of a function is a fundamental concept that ensures the function does not produce any 'jumps' or 'gaps' when its graph is drawn. For a function to be continuous at a point, the function's value at that point must approach the same number from both the left and the right side as we zoom in infinitely close. This is visualized in the study of limits and can be quantified using the epsilon-delta definition.

Epsilon-Delta Definition

The epsilon-delta definition is a formal mathematical description of continuity and is a cornerstone of Real Analysis. This definition underpins the concept of a limit, stating that a function f is continuous at a point x=c if, for every \( \) epsilon (\(\epsilon\)) > 0, there exists a \( \) delta (\(\delta\)) > 0 such that for all x in the domain of f,|x - c| < \(\delta\) implies |f(x) - f(c)| < \(\epsilon\).

This definition characterizes continuity at a single point and encapsulates the idea that small changes in the input near c result in small changes in the output near f(c). In essence, nearly imperceptible variations in x close to c should yield almost indistinguishable variations in f(x). The epsilon-delta definition sets the stage for the more advanced concept of uniform continuity, which addresses continuity across an entire interval or set rather than at a single point.

Uniform Continuity Proof

Uniform continuity is a stronger form of continuity that does not depend on the location within the domain. A function \(f: \textbf{R}^n \rightarrow \textbf{R}^m\) is uniformly continuous on a set S if for every \(\epsilon > 0\), there is a \(\delta > 0\) such that for all \(\textbf{X}, \textbf{Y} \) in S, |\(\textbf{X} - \textbf{Y}\)| < \(\delta\) implies |\(f(\textbf{X}) - f(\textbf{Y})\)| < \(\epsilon\). What makes it uniform is that the same \(\delta\) works for every choice of \(\textbf{X}, \textbf{Y}\) in S.

In proving uniform continuity, it's advantageous to employ the epsilon-delta relationship in a way that covers the entire domain of the function. This frequently involves demonstrating that, no matter how closely you examine the function, you can always find a \(\delta\) such that the change in the function's output is less than the arbitrary \(\epsilon\) you have chosen, regardless of where you are within the domain. Such arguments often include leveraging inequalities, such as the triangle inequality, to show that function components adhere to this uniform behavior. The use of the triangle inequality in vector spaces is an excellent tool for such proofs as it allows bounding the norm of the difference between vectors, leading to the needed conclusion.