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(Q5) Your acquaintance Dr Zookh is very excited because she believes that she has proved the following result. Theorem (Zookh's Boundedness Theorem). Every continuous function f: R+R is bounded. That is, the set {f(x): r ER} is bounded above and below. Proof. Let f: R→ R be continuous. Then f is continuous at 0. Denote the value of f(0) by y. Since f is continuous at 0, we may let € = 1 and conclude that f(x) - y = f(x)-f(0)| <= 1 for all z € R. Thus y-1

(Q5) Your acquaintance Dr Zookh is very excited because she believes that she has proved the following result. Theorem (Zookh's Boundedness Theorem). Every continuous function f: R+R is bounded. That is, the set {f(x): r ER} is bounded above and below. Proof. Let f: R→ R be continuous. Then f is continuous at 0. Denote the value of f(0) by y. Since f is continuous at 0, we may let € = 1 and conclude that f(x) - y = f(x)-f(0)| <= 1 for all z € R. Thus y-1

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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(Q5) Your acquaintance Dr Zookh is very excited because she believes that she has proved the following result. Theorem (Zookh's Boundedness Theorem). Every continuous function f: R → R is bounded. That is, the set {f(x): r ER} is bounded above and below. Proof. Let f: R→ R be continuous. Then f is continuous at 0. Denote the value of f(0) by y. Since f is continuous at 0, we may let € = 1 and conclude that f(x) - y = f(a) f(0)| <= 1 for all x € R. Thus y-1<f(x) <y+1 and so f is bounded below by y-1 and above be y + 1. 0 (a) By giving a counter-example, show that Dr Zookh's "Theorem" is in fact false. (b) In a few sentences, explain in your own words where Dr Zookh's proof goes wrong and how. Your explanation should be written in a way that would be helpful for Dr Zookh to improve her understanding of Real Analysis.
Transcribed Image Text:(Q5) Your acquaintance Dr Zookh is very excited because she believes that she has proved the following result. Theorem (Zookh's Boundedness Theorem). Every continuous function f: R → R is bounded. That is, the set {f(x): r ER} is bounded above and below. Proof. Let f: R→ R be continuous. Then f is continuous at 0. Denote the value of f(0) by y. Since f is continuous at 0, we may let € = 1 and conclude that f(x) - y = f(a) f(0)| <= 1 for all x € R. Thus y-1<f(x) <y+1 and so f is bounded below by y-1 and above be y + 1. 0 (a) By giving a counter-example, show that Dr Zookh's "Theorem" is in fact false. (b) In a few sentences, explain in your own words where Dr Zookh's proof goes wrong and how. Your explanation should be written in a way that would be helpful for Dr Zookh to improve her understanding of Real Analysis.
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