Problem 17 - Basic Analysis: Introduction to Real Analysis

Question

Suppose \(S \subset \mathbb{R}\) and let \(f: S \rightarrow \mathbb{R}\) and \(g: S \rightarrow \mathbb{R}\) be continuous functions. Define \(p: S \rightarrow \mathbb{R}\) by \(p(x):=\max \\{f(x), g(x)\\}\) and \(q: S \rightarrow \mathbb{R}\) by \(q(x):=\min \\{f(x), g(x)\\} .\) Prove that \(p\) and \(q\) are continuous.

Step-by-Step Solution

Answer

Both functions \(p\) and \(q\) are continuous on \(S\).

Step 1: Understand the Definitions

Identify that we have two functions, \(f\) and \(g\), which are continuous on a set \(S\). The functions \(p(x)\) and \(q(x)\) are defined as \(p(x) = \max\{f(x), g(x)\}\) and \(q(x) = \min\{f(x), g(x)\}\) respectively. Our objective is to prove that both \(p\) and \(q\) are continuous functions.

Step 2: Recall Basics of Continuity

Remember that a function \(h: S \to \mathbb{R}\) is continuous at a point \(a \in S\) if \(\lim_{x \to a} h(x) = h(a)\). Since \(f\) and \(g\) are given as continuous, they satisfy this condition at every point \(a \in S\).

Step 3: Formulate Function Inequalities

Express \(p(x)\) and \(q(x)\) using inequalities:- \(p(x) = f(x)\) if \(f(x) \geq g(x)\) and \(p(x) = g(x)\) if \(g(x) > f(x)\).- \(q(x) = f(x)\) if \(f(x) \leq g(x)\) and \(q(x) = g(x)\) if \(g(x) < f(x)\).These inequalities show that near any point, \(p(x)\) and \(q(x)\) switch between \(f(x)\) and \(g(x)\) based on their values.

Step 4: Use Epsilon-Delta Definition

Given \(\epsilon > 0\), since \(f\) and \(g\) are continuous, for any \(a \in S\), there exists \(\delta > 0\) such that for all \(x\) with \(|x-a| < \delta\), we have \(|f(x)-f(a)| < \epsilon\) and \(|g(x)-g(a)| < \epsilon\).

Step 5: Prove Continuity of p(x)

Given any \(a \in S\), choose \(\delta\) as the minimum of the \(\delta\) corresponding to \(f(x)\) and \(g(x)\) ensuring both are within \(\epsilon\). Then: \(\lim_{x \to a} p(x) = \lim_{x \to a} \max(f(x), g(x)) = \max(\lim_{x \to a} f(x), \lim_{x \to a} g(x)) = \max(f(a), g(a)) = p(a)\). Thus, \(p(x)\) is continuous at \(a\).

Step 6: Prove Continuity of q(x)

Similar to \(p(x)\), for \(q(x)\) at \(a \in S\), take \(\delta\) small enough for both \(f(x)\) and \(g(x)\). Then: \(\lim_{x \to a} q(x) = \lim_{x \to a} \min(f(x), g(x)) = \min(\lim_{x \to a} f(x), \lim_{x \to a} g(x)) = \min(f(a), g(a)) = q(a)\). Hence, \(q(x)\) is continuous at \(a\).

Step 7: Conclude Continuity

Since \(a\) was arbitrary and \(p(x)\) and \(q(x)\) satisfy the definition of continuity at every point \(a \in S\), both \(p\) and \(q\) are continuous functions on \(S\).

Key Concepts

Continuous Functions

Continuous functions are an essential concept in calculus and real analysis. At their core, a function is continuous if you can draw its graph without picking up your pencil. Formalized mathematically, a function \(f: S \rightarrow \mathbb{R}\) is continuous at a point \(a \in S\) if the limit of \(f(x)\) as \(x\) approaches \(a\) is equal to \(f(a)\). This means that there are no breaks, jumps, or holes in the function at that point.

Some important properties of continuous functions include:

• The sum or difference of continuous functions is continuous.
• The product of continuous functions is continuous.
• Continuous functions can be neatly described by limits.

Understanding continuous functions is key to tackling more advanced calculus problems, and they serve as the foundation for a lot of mathematical analysis. In this exercise, functions \(f\) and \(g\) are provided as continuous, which serves as the basis for proving new continuous functions \(p\) and \(q\).

Epsilon-Delta Definition

The Epsilon-Delta definition is a rigorous way of defining continuity. It uses the Greek letters \(\epsilon\) (epsilon) and \(\delta\) (delta) to make precise when a function is continuous, especially at a particular point. For \(f: S \rightarrow \mathbb{R}\), \(f\) is continuous at \(a \in S\) if for any small \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever the distance from \(x\) to \(a\) is less than \(\delta\) (i.e., \(|x-a| < \delta\)), the distance from \(f(x)\) to \(f(a)\) is less than \(\epsilon\) (i.e., \(|f(x)-f(a)| < \epsilon\)).

This definition essentially states that we can make \(f(x)\) as close to \(f(a)\) as we want by choosing \(x\) sufficiently close to \(a\). In the given solution, this framework is applied to show how the maximum and minimum of two continuous functions, \(p(x)\) and \(q(x)\), remain continuous. This approach provides a clear pathway to consider continuity for defined functions in complex scenarios.

Limits in Real Analysis

Limits are a cornerstone of calculus and real analysis, allowing us to understand how functions behave as inputs approach specific values. The concept of a limit addresses what value a function approaches as the input approaches some point, even if the function is not defined at that point.

In formal notation, \(\lim_{x \to a} f(x) = L\) means that as \(x\) gets arbitrarily close to \(a\) (but not equal to \(a\)), \(f(x)\) gets arbitrarily close to \(L\).
Understanding limits helps us comprehend continuity in a formal manner, as continuity at a point requires that the limit equals the function's actual value at that point.

In the problem solving approach provided, limits are used to express how \(p(x)\) and \(q(x)\)—functions derived from \(f(x)\) and \(g(x)\)—behave around any point \(a\) in the set \(S\). This is crucial for demonstrating that these functions do not introduce discontinuities. By showing that the limits of \(p(x)\) and \(q(x)\) equal their values at \(a\), the solution confirms that these functions remain smooth, seamlessly inheriting the continuity of \(f\) and \(g\).