s090410 - Math 312, Intro. to Real Analysis: Homework #6...

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Unformatted text preview: Math 312, Intro. to Real Analysis: Homework #6 Solutions Stephen G. Simpson Friday, April 10, 2009 The assignment consists of Exercises 17.3(a,b,c,f), 17.4, 17.9(c,d), 17.10(a,b), 17.14, 18.5, 18.7, 19.1, 19.2(b,c), 19.5 in the Ross textbook. Each exercise counts 10 points. 17.3. (a) By Theorem 17.4(ii) applied three times, cos 4 x is continuous. Then, by Theorem 17.4(i), 1 + cos 4 x is continuous. Then, by Theorem 17.5, log e (1+cos 4 x ) is continuous. Since 1+cos 4 x 1 for all x , the domain of this function is the entire real line. (b) Since x = e log x , it is clear by Theorem 17.4 that x is continuous. It then follows as usual that (sin 2 x + cos 6 x ) is continuous. Again, the domain of this function is the entire real line. (c) We have 2 x 2 = e x 2 log 2 so as before this is continuous. Again, the domain is the entire real line. (f) By Theorem 17.4(iii), 1 /x is continuous for all x negationslash = 0. It then follows as usual that x sin(1 /x ) is continuous for all x negationslash = 0. 17.4. Example 5 in 8 says that if lim x n = x and x n 0 for all n , then lim x n = x . According to Definition 17.1 this says precisely that x is continuous for all x 0. 17.9. (c) Let f ( x ) = x sin(1 /x ) for x negationslash = 0, and let f (0) = 0. We wish to show that f is continuous at 0 using the - definition of continuity. Given > 0, we must find > 0 such that | f ( x ) | < whenever | x | < . We have | f ( x ) | = | x | | sin(1 /x ) | | x | since | sin y | 1 for all y . So let = . Then clearly....
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s090410 - Math 312, Intro. to Real Analysis: Homework #6...

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