7. A function f : (a,6) → R satisfies a Lipschitz condition at x € (a,6) iff there is M > 0 and e > 0 such that |x – y| < e and y E (a,b) imply that |f (x) – f(y)| < M |x – y|. Give an example of a function that fails to satisfy a Lipschitz condition at a point of continuity. If f is differentiable at x, prove that f satisfies a Lipschitz condition at x.

7. A function f : (a,6) → R satisfies a Lipschitz condition at x € (a,6) iff there is M > 0 and e > 0 such that |x – y| < e and y E (a,b) imply that |f (x) – f(y)| < M |x – y|. Give an example of a function that fails to satisfy a Lipschitz condition at a point of continuity. If f is differentiable at x, prove that f satisfies a Lipschitz condition at x.

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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