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3. Find two real-valued functions f and g that are uniformly continuous on a set E, but such that their product fg is not uniformly continuous on E.

3. Find two real-valued functions f and g that are uniformly continuous on a set E, but such that their product fg is not uniformly continuous on E.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Using Example in photo1, solve the same way showing sidework(rightside) and proof(leftside) of the problem in photo2

The image contains a handwritten proof regarding the uniform continuity of a function. The function \( f: [1, \infty) \to \mathbb{R} \) is defined as \( f(x) = \frac{1}{x} \). The goal is to show that \( f \) is uniformly continuous on the interval \([1, \infty)\). ### Proof Outline: 1. **Function Definition and Goal:** - Given \( f(x) = \frac{1}{x} \). - Aim: Prove \( f \) is uniformly continuous on \([1, \infty)\). 2. **Uniform Continuity Definition:** - For all \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( |x - a| < \delta \), then \( |f(x) - f(a)| < \epsilon \). 3. **Proof Details:** - Assume \( \epsilon > 0 \). - Choose \(\delta \leq \epsilon\). - For \( x, a \in [1, \infty) \), if \(|x - a| < \delta\), then: 4. **Calculation of \(|f(x) - f(a)|\):** - \(|f(x) - f(a)| = \left|\frac{1}{x} - \frac{1}{a}\right|\). - Simplification leads to \(\left|\frac{a - x}{xa}\right|\). - Further simplifies to \(\frac{|x - a|}{|x||a|}\). 5. **Inequality Establishment:** - As \(x \geq 1\) and \(a \geq 1\), \(|x||a| \geq 1\). - Therefore, \(\left|\frac{x - a}{x a}\right| \leq |x - a|\). 6. **Conclusion:** - Since \(|x - a| < \delta \leq \epsilon\), - Thus, \(|f(x) - f(a)| \leq \epsilon\). 7. **Final Assertion:** - \( f \) is uniformly continuous on \([1, \infty)\). ### Key Observations: - The proof is structured with
Transcribed Image Text:The image contains a handwritten proof regarding the uniform continuity of a function. The function \( f: [1, \infty) \to \mathbb{R} \) is defined as \( f(x) = \frac{1}{x} \). The goal is to show that \( f \) is uniformly continuous on the interval \([1, \infty)\). ### Proof Outline: 1. **Function Definition and Goal:** - Given \( f(x) = \frac{1}{x} \). - Aim: Prove \( f \) is uniformly continuous on \([1, \infty)\). 2. **Uniform Continuity Definition:** - For all \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( |x - a| < \delta \), then \( |f(x) - f(a)| < \epsilon \). 3. **Proof Details:** - Assume \( \epsilon > 0 \). - Choose \(\delta \leq \epsilon\). - For \( x, a \in [1, \infty) \), if \(|x - a| < \delta\), then: 4. **Calculation of \(|f(x) - f(a)|\):** - \(|f(x) - f(a)| = \left|\frac{1}{x} - \frac{1}{a}\right|\). - Simplification leads to \(\left|\frac{a - x}{xa}\right|\). - Further simplifies to \(\frac{|x - a|}{|x||a|}\). 5. **Inequality Establishment:** - As \(x \geq 1\) and \(a \geq 1\), \(|x||a| \geq 1\). - Therefore, \(\left|\frac{x - a}{x a}\right| \leq |x - a|\). 6. **Conclusion:** - Since \(|x - a| < \delta \leq \epsilon\), - Thus, \(|f(x) - f(a)| \leq \epsilon\). 7. **Final Assertion:** - \( f \) is uniformly continuous on \([1, \infty)\). ### Key Observations: - The proof is structured with
**Problem:** 3. Find two real-valued functions \( f \) and \( g \) that are uniformly continuous on a set \( E \), but such that their product \( fg \) is not uniformly continuous on \( E \). **Explanation:** This problem requires identifying functions \( f \) and \( g \) which, although individually uniformly continuous over the set \( E \), have a product \( fg \) that lacks this property. Uniform continuity is a stricter form of continuity that ensures functions remain continuous regardless of how values approach infinity within the set \( E \). The challenge lies in the interaction between these functions causing the product to lose this uniform continuity, typically due to behaviors like unbounded growth or oscillation.
Transcribed Image Text:**Problem:** 3. Find two real-valued functions \( f \) and \( g \) that are uniformly continuous on a set \( E \), but such that their product \( fg \) is not uniformly continuous on \( E \). **Explanation:** This problem requires identifying functions \( f \) and \( g \) which, although individually uniformly continuous over the set \( E \), have a product \( fg \) that lacks this property. Uniform continuity is a stricter form of continuity that ensures functions remain continuous regardless of how values approach infinity within the set \( E \). The challenge lies in the interaction between these functions causing the product to lose this uniform continuity, typically due to behaviors like unbounded growth or oscillation.
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