Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Problem #100
9:56 1 Safari RealAnalysis-ISBN-fix... Problem 100. Use the definition of continuity to show that if m and b are fixed (but unspecified) real numbers then the function f(x) = mx +b is continuous at every real number a. CONTINUITY: WHAT IT ISN'T AND WHAT It Is 106 Example 10. Use the definition of continuity to show that f(x) = r² is con- tinuous at a = 0. Proof: Let e > 0. Let 8 = VE. If | x – 0| < 8, then |r| < VE. Thus |2² – 0°| = | æ|² < (VE) = e. Thus by the definition, f is continuous at 0. Notice that in these proofs, the challenge of an e > 0 was first given. This is because the choice of & must depend upon ɛ. Also notice that there was no explanation for our choice of d. We just supplied it and showed that it worked. As long as & > 0, then this is all that is required. In point of fact, the d we chose in each example was not the only choice that worked; any smaller å would work as well. Problem 101. (a) Given a particular e > 0 in the definition of continuity, show that if a particular do > 0 satisfies the definition, then any 8 with 0 < ô < ôo will also work for this e. (b) Show that if a d can be found to satisfy the conditions of the definition of continuity for a particular eo > 0, then this & will also work for any e with 0< €o <E. Next Dashboard Calendar To Do Notifications Inbox 因
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Transcribed Image Text:9:56 1 Safari RealAnalysis-ISBN-fix... Problem 100. Use the definition of continuity to show that if m and b are fixed (but unspecified) real numbers then the function f(x) = mx +b is continuous at every real number a. CONTINUITY: WHAT IT ISN'T AND WHAT It Is 106 Example 10. Use the definition of continuity to show that f(x) = r² is con- tinuous at a = 0. Proof: Let e > 0. Let 8 = VE. If | x – 0| < 8, then |r| < VE. Thus |2² – 0°| = | æ|² < (VE) = e. Thus by the definition, f is continuous at 0. Notice that in these proofs, the challenge of an e > 0 was first given. This is because the choice of & must depend upon ɛ. Also notice that there was no explanation for our choice of d. We just supplied it and showed that it worked. As long as & > 0, then this is all that is required. In point of fact, the d we chose in each example was not the only choice that worked; any smaller å would work as well. Problem 101. (a) Given a particular e > 0 in the definition of continuity, show that if a particular do > 0 satisfies the definition, then any 8 with 0 < ô < ôo will also work for this e. (b) Show that if a d can be found to satisfy the conditions of the definition of continuity for a particular eo > 0, then this & will also work for any e with 0< €o <E. Next Dashboard Calendar To Do Notifications Inbox 因
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