1873_solutions

2 and so the proof is complete 5 a a function f is

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Unformatted text preview: e that the limit of x n belongs to S. In view of Part a, the present result follows at once from an earlier theorem on limits of sequences. d. Suppose that f is uniformly continuous on a set S, that x is a real number and that x n and t n are sequences in S that converge to the number x. Prove that lim lim f x n  n Ý f t n . nÝ The existence of these limits was guaranteed in Part c. Now since t n x n 0 as n Ý we deduce from the relationship between uniform continuity and limits of sequences that f xn 0 as n Ý. f tn e. Suppose that f is uniformly continuous on a set S and that x S S. Explain how we can use Part d to extend the definition of the function f to the number x in such a way that f is continuous on the set SÞ x . We know that there exists a sequence x n in S that converges to x and we know that there is a single limit for all of the sequences f x n that can be made in this way. We define f x to be this common limit value. This extension of the function f to the set S Þ x is uniformly continuous. The proof will be given in the more extended case that we consider below in Part f. f. Prove that if f is uniformly continuous on a set S then it is possible to extend f to the closure S of S in such a way that f is uniformly continuous on S. For every number x S S we define f x by the method described in Part e. To show that the extension of f to S is uniformly continuous, suppose that  0. Using the uniform continuity of f on S we choose   0 such that the inequality |f t f x |  2 holds whenever t and x belong to S and |t x |  . We shall now observe that whenever t and x belong to S and |t x |   we must have |f t f x |  . To make this observation, suppose that t and x belong to S and that |t x |  . choose a sequence t n in S that converges to t and a sequence x n in S that converges to x. Since lim |t x |  n Ý|t n x n | we know that the inequality |t n x n |   must hold for all n sufficiently large and therefore, since 204 |f t f x |  n Ý|f t n lim f xn | and since |f t n for all n sufficiently large we must have |f t f xn |  fx | 2 2 . 10. Suppose that f is a continuous function on a bounded set S. rove that the following two conditions are equivalent: a. The function f is uniformly continuous on S. b. It is possible to extend f to a continuous function on the set S. The fact that condition a implies condition b follows from Exercise 9. On the other hand, if f has a continuous extension to the set S then, this extension, being a continuous function on a closed bounded set, must be uniformly continuous; and so f must be uniformly continuous on S. 11. Given that f is a function defined on a set S of real numbers, prove that the following conditions are equivalent: a. The function f fails to be uniformly continuous on the set S. b. There exists a number n Ý and  0 and there exist two sequences t n and x n in S such that t n |f x n xn 0 as f tn | for every n. At the suggestion of my good friend Sean Ellermeyer this exerci...
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