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Let A:= (0, oo) and let k: A-+ R be defined as follows. For x e A, x irrational, we

Introduction to Real Analysis | 3rd Edition | ISBN: 9780471321484 | Authors: Robert G. Bartle, Donald R. Sherbert ISBN: 9780471321484 424

Solution for problem 14 Chapter 5.1

Introduction to Real Analysis | 3rd Edition

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Introduction to Real Analysis | 3rd Edition | ISBN: 9780471321484 | Authors: Robert G. Bartle, Donald R. Sherbert

Introduction to Real Analysis | 3rd Edition

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Problem 14

Let A:= (0, oo) and let k: A-+ R be defined as follows. For x e A, x irrational, we define k(x) = 0; for x e A rational and of the form x = m/n with natural numbers m, n having no common factors except I, we define k(x) := n. Prove that k is unbounded on every open interval in A. Conclude that k is not continuous at any point of A. (See Example 5.1.6(h).)

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Chapter 5.1, Problem 14 is Solved
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Textbook: Introduction to Real Analysis
Edition: 3
Author: Robert G. Bartle, Donald R. Sherbert
ISBN: 9780471321484

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Let A:= (0, oo) and let k: A-+ R be defined as follows. For x e A, x irrational, we

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