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Suppose DCR and f.D R and g:D-R are both uniformly continuous on D. (a) Prove that the function f + g: D-R is uniformly continuous on D. (b) Give a counterexample to show that the function fg: D-R need not be uniformly continuous on D.

Suppose DCR and f.D R and g:D-R are both uniformly continuous on D. (a) Prove that the function f + g: D-R is uniformly continuous on D. (b) Give a counterexample to show that the function fg: D-R need not be uniformly continuous on D.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.5: Rational Functions
Problem 50E
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## Uniform Continuity Proof and Counterexample ### Statement: Suppose \( D \subseteq \mathbb{R} \) and functions \( f: D \to \mathbb{R} \) and \( g: D \to \mathbb{R} \) are both uniformly continuous on \( D \). #### (a) Prove that the function \( f + g: D \to \mathbb{R} \) is uniformly continuous on \( D \). #### (b) Provide a counterexample to show that the function \( f \cdot g: D \to \mathbb{R} \) need not be uniformly continuous on \( D \). ### Explanation: - **Uniform Continuity**: A function \( h: D \to \mathbb{R} \) is uniformly continuous on \( D \) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x, y \in D \), \( |x - y| < \delta \) implies \( |h(x) - h(y)| < \epsilon \). ### Tasks: 1. **Proof of Uniform Continuity for Sum of Functions**: - To prove \( f + g \) is uniformly continuous on \( D \), utilize the uniform continuity of \( f \) and \( g \) on \( D \). 2. **Counterexample for Product of Functions**: - Find functions \( f \) and \( g \) that are uniformly continuous on \( D \), but whose product \( f \cdot g \) is not uniformly continuous on \( D \). ### Analysis for Educators: - Encourage students to review the definitions and properties of uniform continuity. - Discuss the implications of uniform continuity in analysis and real-world problems. - Guide students through the proof technique required for demonstrating uniform continuity of the sum of functions. - Illustrate with examples and encourage students to create their own counterexamples for better understanding. ### Graph Explanation (if provided): If graphs or diagrams were to be present, they would typically illustrate the behavior of functions \( f \) and \( g \) and their sum \( f + g \), showing how their slopes or values remain bounded, underlining the concept of uniform continuity visually. For counterexamples, graphs would show instances where the product \( f \cdot g \) shows non-uniform characteristics.
Transcribed Image Text:## Uniform Continuity Proof and Counterexample ### Statement: Suppose \( D \subseteq \mathbb{R} \) and functions \( f: D \to \mathbb{R} \) and \( g: D \to \mathbb{R} \) are both uniformly continuous on \( D \). #### (a) Prove that the function \( f + g: D \to \mathbb{R} \) is uniformly continuous on \( D \). #### (b) Provide a counterexample to show that the function \( f \cdot g: D \to \mathbb{R} \) need not be uniformly continuous on \( D \). ### Explanation: - **Uniform Continuity**: A function \( h: D \to \mathbb{R} \) is uniformly continuous on \( D \) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x, y \in D \), \( |x - y| < \delta \) implies \( |h(x) - h(y)| < \epsilon \). ### Tasks: 1. **Proof of Uniform Continuity for Sum of Functions**: - To prove \( f + g \) is uniformly continuous on \( D \), utilize the uniform continuity of \( f \) and \( g \) on \( D \). 2. **Counterexample for Product of Functions**: - Find functions \( f \) and \( g \) that are uniformly continuous on \( D \), but whose product \( f \cdot g \) is not uniformly continuous on \( D \). ### Analysis for Educators: - Encourage students to review the definitions and properties of uniform continuity. - Discuss the implications of uniform continuity in analysis and real-world problems. - Guide students through the proof technique required for demonstrating uniform continuity of the sum of functions. - Illustrate with examples and encourage students to create their own counterexamples for better understanding. ### Graph Explanation (if provided): If graphs or diagrams were to be present, they would typically illustrate the behavior of functions \( f \) and \( g \) and their sum \( f + g \), showing how their slopes or values remain bounded, underlining the concept of uniform continuity visually. For counterexamples, graphs would show instances where the product \( f \cdot g \) shows non-uniform characteristics.
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