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A function f : D → R is called a Lipschitz function if there is a number K > 0 such that |f(x) – f(y)| < K|x – y| for all æ and y in the domain D. (a) Prove that a Lipschitz function satisfies the e-ô criterion on its domain, and the conclude that it is uniformly continuous. (b) Let f(x) = Va for all a e (0, 1). Prove that f is uniformly continuous but is not a Lipschitz function.

A function f : D → R is called a Lipschitz function if there is a number K > 0 such that |f(x) – f(y)| < K|x – y| for all æ and y in the domain D. (a) Prove that a Lipschitz function satisfies the e-ô criterion on its domain, and the conclude that it is uniformly continuous. (b) Let f(x) = Va for all a e (0, 1). Prove that f is uniformly continuous but is not a Lipschitz function.

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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A function f : D → R is called a Lipschitz function if there is a number K > 0 such that |f(x) – f(y)| < K|x – y| for all æ and y in the domain D. (a) Prove that a Lipschitz function satisfies the e-ô criterion on its domain, and the conclude that it is uniformly continuous. (b) Let f(x) = Va for all a e (0, 1). Prove that f is uniformly continuous but is not a Lipschitz function.
Transcribed Image Text:A function f : D → R is called a Lipschitz function if there is a number K > 0 such that |f(x) – f(y)| < K|x – y| for all æ and y in the domain D. (a) Prove that a Lipschitz function satisfies the e-ô criterion on its domain, and the conclude that it is uniformly continuous. (b) Let f(x) = Va for all a e (0, 1). Prove that f is uniformly continuous but is not a Lipschitz function.
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