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Exercise 1. Recall that a function f: ER is called Lipschitz if there exists M such that |f(x) = f(y)| ≤ M|x - y| holds for all x, y ≤ E. i) Prove that if f: ER is Lipschitz, then f(x) is uniformly continuous. ii) Prove that f: [a, co) → R given by x√√x is Lipschitz on iii) Show f: [0, 0)→ R given by x√x is uniformly continuous, but not Lipschitz. [a, ∞o) for all a>0.

Exercise 1. Recall that a function f: ER is called Lipschitz if there exists M such that |f(x) = f(y)| ≤ M|x - y| holds for all x, y ≤ E. i) Prove that if f: ER is Lipschitz, then f(x) is uniformly continuous. ii) Prove that f: [a, co) → R given by x√√x is Lipschitz on iii) Show f: [0, 0)→ R given by x√x is uniformly continuous, but not Lipschitz. [a, ∞o) for all a>0.

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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Exercise 1. Recall that a function f: ER is called Lipschitz if there exists M such that |f(x) = f(y)| ≤ M/x-y| holds for all x, y EE. i) Prove that if f: ER is Lipschitz, then f(x) is uniformly continuous. ii) Prove that f: [a, ∞o) → R given by x√x is Lipschitz on [a, co) for all a > 0. iii) Show f: [0, ∞) → R given by x√x is uniformly continuous, but not Lipschitz.
Transcribed Image Text:Exercise 1. Recall that a function f: ER is called Lipschitz if there exists M such that |f(x) = f(y)| ≤ M/x-y| holds for all x, y EE. i) Prove that if f: ER is Lipschitz, then f(x) is uniformly continuous. ii) Prove that f: [a, ∞o) → R given by x√x is Lipschitz on [a, co) for all a > 0. iii) Show f: [0, ∞) → R given by x√x is uniformly continuous, but not Lipschitz.
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