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Use Theorem 9.2.1 to show that T is not continuous at 0, no matter what value is chosen for b.

Use Theorem 9.2.1 to show that T is not continuous at 0, no matter what value is chosen for b.

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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12:20 M Use Theorem 9.2.1 to show that T is not continuous at 0, no matter what value is chosen for b. Theorem 9.2.1. The function f is continuous at a if and only if f satisfies the following property: V sequences (xn), if lim n = a then lii in-context Sketch of the Proof of Theorem 9.2.1. We've seen how we can use Theorem 9.2.1, now we need to prove Theorem 9.2.1. The forward direction is fairly straightforward. So we assume that f is continuous at a and start with a sequence (xn) which converges to a. What is left to show is that lim,→0 f(xn) = f(a). If you write down the definitions of f being continuous at a, lim,- Xn = a, and lim, +00 f(xn) = f(a), you should be able to get from what you are assuming to what you want to conclude. To prove the converse, it is convenient to prove its contrapositive. That is, we want to prove that if ƒ is not continuous at a then we can construct a sequence (x.) > II
Transcribed Image Text:12:20 M Use Theorem 9.2.1 to show that T is not continuous at 0, no matter what value is chosen for b. Theorem 9.2.1. The function f is continuous at a if and only if f satisfies the following property: V sequences (xn), if lim n = a then lii in-context Sketch of the Proof of Theorem 9.2.1. We've seen how we can use Theorem 9.2.1, now we need to prove Theorem 9.2.1. The forward direction is fairly straightforward. So we assume that f is continuous at a and start with a sequence (xn) which converges to a. What is left to show is that lim,→0 f(xn) = f(a). If you write down the definitions of f being continuous at a, lim,- Xn = a, and lim, +00 f(xn) = f(a), you should be able to get from what you are assuming to what you want to conclude. To prove the converse, it is convenient to prove its contrapositive. That is, we want to prove that if ƒ is not continuous at a then we can construct a sequence (x.) > II
12:19 M Problem 9.2.5. The function T(x) = sin (-) is often called the topologist's sine curve. Whereas sin x has roots at nr, n E Z and oscillates infinitely often as x → ±∞, T has roots at n E Z, n # 0, and oscillates infinitely often as x approaches zero. A rendition of the graph follows. -0.5 0.5 15 Notice that T is not even defined at = 0. We can extend T to be defined at 0 by simply choosing a value for T(0) : sin (+), if x 70 16, T(x) = if x = Use Theorem 9.2.1 to show that T is not continuous at 0, no matter what value is chosen for b. II
Transcribed Image Text:12:19 M Problem 9.2.5. The function T(x) = sin (-) is often called the topologist's sine curve. Whereas sin x has roots at nr, n E Z and oscillates infinitely often as x → ±∞, T has roots at n E Z, n # 0, and oscillates infinitely often as x approaches zero. A rendition of the graph follows. -0.5 0.5 15 Notice that T is not even defined at = 0. We can extend T to be defined at 0 by simply choosing a value for T(0) : sin (+), if x 70 16, T(x) = if x = Use Theorem 9.2.1 to show that T is not continuous at 0, no matter what value is chosen for b. II
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