:Theorem 4.4.5 (Sequential Criterion for Absence of Uniform Conti-nuity). A function f AR fails to be uniformly continuous on A if andonly if there exists a particular eo > 0 and two sequences (xn) and (yn) in Asatisfyingxn - Yn → 0but|f(xn) — f(yn)|≥ €0. Exercise 4.4.1. (a) Show that f(x) = x³ is continuous on all of R.(b) Argue, using Theorem 4.4.5, that f is not uniformly continuous on R.(c) Show that f is uniformly continuous on any bounded subset of R.

Answered: Image /qna-images/answer/97c52df9-e0ea-4811-9ba4-fc41933ab89b.jpg

Answered: : Theorem 4.4.5 (Sequential Criterion…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 21E: [Type here] 21. Prove that ifand are integral domains, then the direct sum is not an integral...

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