Using Example in photo1, solve the same way showing sidework(rightside) and proof(leftside) of the problem in photo2 **Problem 4**Choose and work either a) or b), NOT BOTH.(a) Show that the function \( f(x) = \sin x \) for \( x \in \mathbb{R} \) is uniformly continuous.(b) Show that the function \( k(x) = \sin(1/x) \) is not uniformly continuous on the set \( D(k) = \{ x \in \mathbb{R}^+ : 0 < x \} \). The image contains handwritten mathematical notes related to proving the uniform continuity of a function on an interval. It deals with the function \( f(x) = \frac{1}{x} \) on the interval \([1, \infty)\).### Content Details:1. **Function Definition:** - \( f: [1, \infty) \rightarrow \mathbb{R}, \, f(x) = \frac{1}{x} \).2. **Goal:** - The objective is to prove that \( f \) is uniformly continuous on \([1, \infty)\).3. **Uniform Continuity Definition:** - For every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x, u \) in \([1, \infty)\), if \( |x-u| < \delta \), then \( |f(x) - f(u)| < \varepsilon \).4. **Proof Structure:** - If \( \varepsilon > 0 \), choose \( \delta = \varepsilon \). - Assume \( x, u \in [1, \infty) \).5. **Evaluation of Difference:** - \(|f(x) - f(u)| = \left|\frac{1}{x} - \frac{1}{u}\right|\) - This simplifies to: \[ \left|\frac{u - x}{x \cdot u}\right| \]6. **Conclusion:** - Based on calculations, \(|x-u| \leq \delta = \varepsilon\) ensures \(|f(x) - f(u)| < \varepsilon\).7. **Additional Conditions:** - \( x \geq 1, u \geq 1 \). - There are bounding conditions discussed with inequalities.### Diagram Explanation:- The notes use colored text and underline to highlight key steps in the derivation.- The bounding of terms like \(|x-u|\) and manipulation of \( \delta \) and \( \varepsilon \) demonstrate methods to show uniform continuity.- The final conclusion reinforces the result that the function is indeed uniformly continuous on the specified interval.

In the question given that we have to solve any one of them. Let us solve the first one.

Answered: 4. Choose and work either a) or b), NOT…

4. Choose and work either a) or b), NOT BOTH. (a) Show that the function f (x) = sin z for x €R is uniformly continuous.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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