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Exercise 3.4.1. Using only the definition of uniform continuity. prove that the fol- lowing functions are uniformly continuous. (1) Let f: [0,3] – R be defined by f(x) =x² for all x € [0,3]. (2) Let g: [1,2) – R be defined by g(x) = Vī for all xE [1,2).

Exercise 3.4.1. Using only the definition of uniform continuity. prove that the fol- lowing functions are uniformly continuous. (1) Let f: [0,3] – R be defined by f(x) =x² for all x € [0,3]. (2) Let g: [1,2) – R be defined by g(x) = Vī for all xE [1,2).

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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Please include a formal proof. Thanks!

**Exercise 3.4.1**: Using only the definition of uniform continuity, prove that the following functions are uniformly continuous. 1. Let \( f: [0,3] \to \mathbb{R} \) be defined by \( f(x) = x^2 \) for all \( x \in [0,3] \). 2. Let \( g: [1,2] \to \mathbb{R} \) be defined by \( g(x) = \sqrt{x} \) for all \( x \in [1,2] \).
Transcribed Image Text:**Exercise 3.4.1**: Using only the definition of uniform continuity, prove that the following functions are uniformly continuous. 1. Let \( f: [0,3] \to \mathbb{R} \) be defined by \( f(x) = x^2 \) for all \( x \in [0,3] \). 2. Let \( g: [1,2] \to \mathbb{R} \) be defined by \( g(x) = \sqrt{x} \) for all \( x \in [1,2] \).
**Definition 3.4.1.** Let \( A \subseteq \mathbb{R} \) be a set, and let \( f : A \rightarrow \mathbb{R} \) be a function. The function \( f \) is uniformly continuous if for each \( \varepsilon > 0 \), there is some \( \delta > 0 \) such that \( x, y \in A \) and \( |x - y| < \delta \) imply \( |f(x) - f(y)| < \varepsilon \). ∎
Transcribed Image Text:**Definition 3.4.1.** Let \( A \subseteq \mathbb{R} \) be a set, and let \( f : A \rightarrow \mathbb{R} \) be a function. The function \( f \) is uniformly continuous if for each \( \varepsilon > 0 \), there is some \( \delta > 0 \) such that \( x, y \in A \) and \( |x - y| < \delta \) imply \( |f(x) - f(y)| < \varepsilon \). ∎
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