Problem 11 - Basic Analysis: Introduction to Real Analysis

Question

Prove: a) If \(f: S \rightarrow \mathbb{R}\) and \(g: S \rightarrow \mathbb{R}\) are uniformly continuous, then \(h: S \rightarrow \mathbb{R}\) given by \(h(x):=f(x)+g(x)\) is uniformly continuous. b) If \(f: S \rightarrow \mathbb{R}\) is uniformly continuous and \(a \in \mathbb{R},\) then \(h: S \rightarrow \mathbb{R}\) given by \(h(x):=a f(x)\) is uniformly continuous.

Step-by-Step Solution

Answer

Both \( h(x) = f(x) + g(x) \) and \( h(x) = a f(x) \) are uniformly continuous given \( f \) and \( g \) are uniformly continuous and \( a \) is a real number.

Step 1: Understanding Uniform Continuity

A function \( f: S \rightarrow \mathbb{R} \) is uniformly continuous if for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x, y \in S \), if \( |x - y| < \delta \) then \( |f(x) - f(y)| < \varepsilon \). This means that the choice of \( \delta \) is independent of the particular points \( x \) and \( y \).

Step 2: Statement for Part a

We need to show that if \( f \) and \( g \) are uniformly continuous, then the function \( h(x) = f(x) + g(x) \) is also uniformly continuous.

Step 3: Uniform Continuity of f and g

Since \( f \) is uniformly continuous, for every \( \varepsilon > 0 \), there exists \( \delta_1 > 0 \) such that \( |f(x) - f(y)| < \varepsilon/2 \) whenever \( |x - y| < \delta_1 \). Similarly, for \( g \), there exists \( \delta_2 > 0 \) such that \( |g(x) - g(y)| < \varepsilon/2 \) whenever \( |x - y| < \delta_2 \).

Step 4: Showing h is Uniformly Continuous

Take \( \delta = \min(\delta_1, \delta_2) \). Then for any \( x, y \in S \) with \( |x - y| < \delta \), we have \(|f(x) - f(y)| < \varepsilon/2\) and \(|g(x) - g(y)| < \varepsilon/2\). Therefore, \(|h(x) - h(y)| = |f(x)+g(x) - (f(y)+g(y))| \leq |f(x)-f(y)| + |g(x)-g(y)| < \varepsilon/2 + \varepsilon/2 = \varepsilon\), showing that \( h \) is uniformly continuous.

Step 5: Statement for Part b

We need to show that if \( f \) is uniformly continuous and \( a \in \mathbb{R} \), then \( h(x) = a f(x) \) is also uniformly continuous.

Step 6: Uniform Continuity Implication

Since \( f \) is uniformly continuous, for every \( \varepsilon > 0 \), there exists \( \delta > 0 \) such that \( |f(x) - f(y)| < \varepsilon/|a| \) for all \( x, y \in S \) with \( |x - y| < \delta \).

Step 7: Showing h(x) = a f(x) is Uniformly Continuous

For any \( x, y \in S \) with \( |x - y| < \delta \), we have \(|h(x) - h(y)| = |a f(x) - a f(y)| = |a||f(x) - f(y)| < |a|(\varepsilon/|a|) = \varepsilon\), thus \( h(x) \) is uniformly continuous.

Key Concepts

Real Analysis

Real Analysis is a branch of mathematics focusing on the behavior of real numbers and real-valued functions. In this context, a critical concept is continuity, particularly uniform continuity.

• Uniform Continuity: This refers to a function where the speed at which changes happen is steady across its entire domain. This is different from just continuity, where the changes could vary at different points.
• Understanding the Domain: Uniform continuity ensures that for all points within a set, certain predictable behaviors happen without having to change parameters for smaller subintervals.

By examining real-valued functions defined over some domain, students begin to appreciate how different kinds of continuity affect function properties and conclusions derived thereafter. In the problem above, uniform continuity is at the heart of proving the given functions' properties for sums and scalar multiples.

Function Properties

Understanding function properties, like continuity, is fundamental in real analysis. One essential property is how functions interact when combined or modified.

• Sums of Functions: When two functions, both uniformly continuous, are added, the resulting function also retains its uniform continuity. This is because the combined fluctuations of both functions still respect a certain uniform behavior controlled by the smallest of their constraints.
• Scaling Functions: When a function is uniformly continuous, and you multiply it by a constant, the resultant function retains the property of uniform continuity. The constant scales the rate of change but does not disturb uniform behavior.

These properties are critical for students learning how function transformations preserve or change certain behaviors like uniform continuity. It contributes to a deeper understanding of how to handle complex functions in analysis.

Mathematical Proofs

Mathematical proofs serve as the foundation for reliably confirming the truth of mathematical statements. They offer conviction beyond intuition by providing a logical framework of argumentation.

• Proof Structure: Typically, proofs start by clearly stating the assumptions, the desired conclusion, and then logically linking the two through sequential reasoning.
• Direct Proofs: The direct method used in demonstrating properties of uniform continuity shows how assumed properties (e.g., continuity of individual components) logically lead to conclusions about composite functions.
• Problem Solving: In the uniform continuity problem, understanding the conditions that ensure a certain behavior helps formulate the steps necessary for the proof. Knowing how to select parameters like \(\delta\) to satisfy \(\varepsilon\) constraints becomes crucial.

Through building proofs for function properties, students gather insights into systematic thinking and develop skills that are applicable in broader realms of analysis or even in other mathematical fields.