Problem 3 - Advanced Calculus

Question

Let \(f\) and \(g\) each be uniformly continuous on a set \(E\). Show that \(f+g\) is uniformly continuous on \(E\).

Step-by-Step Solution

Answer

The function \(f+g\) is uniformly continuous on \(E\).

Step 1: Understand Uniform Continuity

Uniform continuity for a function \(f\) is defined such that for every \(\epsilon > 0\), there exists \(\delta > 0\) with the property that if two elements \(x, y\) in \(E\) satisfy \(|x - y| < \delta\), then \(|f(x) - f(y)| < \epsilon\). This means the difference in the function values is small if the difference in the function inputs is small.

Step 2: Apply Definition of Uniform Continuity to \(f\) and \(g\)

Given that \(f\) and \(g\) are uniformly continuous on \(E\) means that for every \(\epsilon > 0\), there exist \(\delta_f > 0\) and \(\delta_g > 0\) such that if \(x, y\) in \(E\) satisfy \(|x - y| < \delta_f\) and \(|x - y| < \delta_g\), then \(|f(x) - f(y)| < \epsilon/2\) and \(|g(x) - g(y)| < \epsilon/2\).

Step 3: Show that \(f+g\) is Uniformly Continuous

Now we want to show that \(f+g\) is uniformly continuous on \(E\). Let \(\delta = \min(\delta_f, \delta_g)\). If \(|x - y| < \delta, |f(x) - f(y)| < \epsilon/2\) and \(|g(x) - g(y)| < \epsilon/2\). Then, by the triangle inequality, \(|(f+g)(x) - (f+g)(y)| = |f(x) - f(y) + g(x) - g(y)| \leq |f(x) - f(y)| + |g(x) - g(y)| < \epsilon/2 + \epsilon/2 = \epsilon\). This shows that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(x, y\) in \(E\) satisfy \(|x - y| < \delta\), then \(|(f+g)(x) - (f+g)(y)| < \epsilon\). Therefore, \(f+g\) is uniformly continuous on \(E\).

Key Concepts

Triangle Inequality

The Triangle Inequality is a fundamental concept in mathematics that helps us deal with the sum of absolute values. It's especially useful when dealing with uniformly continuous functions, like in the exercise above. The Triangle Inequality states that for any real numbers (or vectors) \(a\) and \(b\), the absolute value of their sum is less than or equal to the sum of their absolute values expressed as:

• \(|a + b| \leq |a| + |b|\)

This was key in showing that \(f+g\) is uniformly continuous. When we use the Triangle Inequality, we break down complex expressions into manageable parts. This allows us to control the total sum of absolute differences between the function values. In this way, if each function part (like \(f(x) - f(y)\) and \(g(x) - g(y)\)) is small, then their entire sum is also small. This property helped demonstrate that the whole function \(f+g\), is within the desired range of variation.

Function Properties

Understanding the basic properties of functions is crucial when dealing with problems like uniform continuity. Every function might behave differently based on certain characteristics:

• Continuity: This is a basic property where small changes in the input result in small changes in the output.
• Uniform Continuity: A stronger version, meaning the same level of continuity holds true throughout the entire domain.
• Linearity of Addition: Functions can be added together (\(f+g\)) easily, and their continuity properties often align.

In the given problem, since each function \(f\) and \(g\) is uniformly continuous, their sum \(f+g\) retains this property. Uniform continuity ensures that the way each function varies is predictably manageable. This aligning property means changes in the input lead to uniformly small changes in the output, thus combining functions preserves this behavior.

Epsilon-Delta Definition

In calculus and analysis, the epsilon-delta (\(\epsilon\) and \(\delta\)) definition is a precise way to define uniform continuity. For a function \(f\) to be uniformly continuous:

• For every \(\epsilon > 0\), there must be a \(\delta > 0\) such that
• For all \(x, y\) in the domain, if \(|x - y| < \delta\), then \(|f(x) - f(y)| < \epsilon\).

In plain terms, this definition means that no matter how small a change we specify in the output (\(\epsilon\)), we can find a threshold (\(\delta\)) to ensure the input stays within this change. This is applied to both functions \(f\) and \(g\) in the solution: by setting \(\epsilon/2\) and choosing \(\delta\) appropriately, the sum \(f+g\) remains uniformly continuous as well. This definition allows us to mathematically assure the desired smallness of changes throughout the function domain and is the backbone of proving uniform continuity in rigorous terms.