1873_solutions

Way in which the integer k was chosen r that f 1 1

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Unformatted text preview: will follow at once from the obvious fact that f x  0 for every number x. Exercises on the Distance Function 1. Two sets A and B of real numbers are said to be separated from each other if A BA B . Prove that if two sets A and B are separated from each other then A x B x  0 whenever x A and A x B x  0 whenever x B. 2. Prove that if two sets A and B are separated from each other then there exist two open sets U and V that are disjoint from each other such that A U and B V. Solution: We define 197 U x R A x B x  0 V x R A x B x  0 . and Since the function  A  B is continuous on R we deduce from the first of some earlier exercises that the sets U and V are open. 3. Given two sets A and B of real numbers, prove that the following conditions are equivalent: a. We have A b. There exists a continuous function f : R B . 0, 1 such that fx  0 if x A 1 if x B . We deduce at once from Urysohn’s lemma that condition a implies condition b. On the other hand, if f is a continuous function that takes the value 0 at every number x A and takes the value 1 at every number x B then 1 A x fx 3 and 2. B x fx 3 4. Suppose that A, B and C are closed sets of real numbers and no two of these three sets intersect. Prove that there exists a continuous function f on R such that 1 if x 2 if x B. 3 if x fx  A C Using Urysohn’s lemma we choose two continuous functions g 1 and g 2 from R into 0, 1 such that g 1 x  0 whenever x A and g 1 x  1 whenever x B Þ C and g 2 x  0 whenever x A Þ B and g 2 x  1 whenever x C. We define g 3 x  1 for every number x. We now define f  g1  g2  g3. 5. Suppose that S is a set of real numbers and that S fails to be closed. Prove that there exists a convergent sequence x n in S and a continuous function f on the set S such that the sequence f x n fails to converge. Solution: We begin by choosing a number w S S and we choose a sequence x n in S that converges to the number w. We define E to be the range of the sequence x n and, Using the fact that that the set E must be infinite, we choose an infinite subset A of E such that the set B  E A is also infinite. Since no member of S can lie in the closures of both A and B, the function  f   A B A is continuous on the set S. Furthermore, since there must be infinitely many integers n for which x n A and infinitely many integers n for which x n B the sequence f x n has no limit. Some Exercises on Continuous Functions on Closed Bounded Sets 1. Give an example of a function f that is continuous on a closed set H such that the range f H of the function f 198 fails to be closed. We can define f x  1 for x x is 0, 1 which is not closed. 1. This function f is continuous on the closed set 1, Ý and its range 2. Give an example of a function f that is continuous on a closed set H such that the range f H of the function f fails to be bounded. We can define f x  x for every x R. 3. Give an example of a function f that is continuous on a bounded set H such that the range f H of t...
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