1873_solutions

F is a bounded subset of y we know that whenever s is

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Unformatted text preview: 1 , x 2   we have . d g f x1 , g f x2 11. Suppose that f is a continuous function on a subset S of a compact metric space X. Prove 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 at once from Exercise 7. Since S is compact, a continuous function on S must be uniformly continuous. Therefore, if f has a continuous extension to S, then f must be uniformly continuous on S. 12. Is it true that if f is a uniformly continuous function from a bounded metric space X onto a metric space Y then Y is bounded? Not a chance! If X is the discrete space in which 0 if x  y d x, y  1 if x y then every function from X to another metric space must be uniformly continuous. 13. Is it true that if f is a uniformly continuous function from a complete metric space X onto a metric space Y then Y is complete? No, this statement is false. For example, if we define 1 fx  1  x2 for x 0, Ý then f is a uniformly continuous function from 0, Ý onto 0, 1 . 14. Suppose that f is a one-one uniformly continuous function from a metric space X onto a metric space Y and that the inverse function f 1 of f is continuous on Y. Prove that if Y is complete then so is X. Suppose that x n is a Cauchy sequence in the space X. Since f is uniformly continuous from X to Y, it is easy to see that the sequence f x n is a Cauchy sequence in the space Y. Since Y is complete, the sequence f x n must converge. We define y  n Ý f xn . lim Since the function f 1 is continuous at the point y we have f 1 f x n f 1 y as n Ý. In other 1 y as n words, x n f Ý and we have shown that the sequence x n is convergent. 15. True or false: If f is a uniformly continuous one-one function from a complete metric space X onto a complete metric space Y then the inverse function f 1 of f must be continuous. No, this statement is false. We take X  0, 1 Þ 2, Ý and we define fx  x 1 235 if 0 1 x x if x 2 1 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 2 3 4 x 5 6 7 8 The above figure also shows the line y  1 in red. We see that f is a uniformly continuous one-one function from the complete space 0, 1 Þ 2, Ý onto the complete space 0, 3 . 2 16. We shall say that a subset S of a metric space X is compressed if for every number different points x and t in S such that d t, x  .  0 there exist two a. Prove that every compressed subset of a metric space must be infinite. This statment is obvious because any finite set positive numbers has a least member that is positive. b. Prove that if a subset S of a metric space X has a limit point then S must be compressed. Suppose that S is a subset of a metric space X and that S has a limit point x. Suppose that  0. Choose a point u B x, /2 x . Now, using the fact that the number d x, u is positive, choose a point v B x, d x, u x . Since both u and v belong to the ball B x, /2 we have d u, v  . c. Give an example of...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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