1873_solutions

Of proof by mathematical induction which is available

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Unformatted text preview: e f H of f fails to be closed in R. We can define f x  1 for x 1. This function f is continuous on the closed set 1, Ý and its range x is 0, 1 which is not closed. 2. Give an example of a function f that is continuous from a closed set H of real numbers into R such that range f H of 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 from a bounded set H of real numbers into R such that range f H of f fails to be closed in R. We can define f x  x for 0  x  1. 4. Give an example of a function f that is continuous from a bounded set H of real numbers into R such that range f H of f fails to be bounded. We can define f x  1 for 0  x  1. x 5. Prove that if a set H of real numbers is unbounded above and f x  x for every number x in H, then f is a continuous function on H and f fails to have a maximum. Since the range of f is the set H which is assumed to be unbounded, the function f must be unbounded above. 6. Prove that if S is a subset of a metric space X and if a is a point of X that is close to S but not a member of S then the function f defined by the equation 1 fx  d a, x for every x X is unbounded.We can see that f has no maximum by showing that f is unbounded above. Given any positive number q the inequality fx q says that 1 q d a, x which holds when d a, x  1 q But, since a is close to the set S, we know that there do indeed exist members x of S for which the inequality 227 a|  1 q holds. Therefore there are members x in H for which f x  q and we have shown that f fails to be bounded above. |x 7. Is it true that if every continuous function from a given metric space X to 0, Ý has a minimum then X must be compact? Yes, it’s true. The theorems of this section tell us that if a metric space X is not compact then there exists a continuous unbounded real function on X. For any such function f, the function 1 g 1  f2 is continuous (and bounded) on X and has no minimum. 8. Is it true that if X is a complete metric space and f is a continuous function from X onto a metric space Y then Y must also be complete? The answer is no. Define fx  1 x for every number x in the complete metric space 1, Ý . The range of f is the metric space 0, 1 , which is not complete. 9. Is it true that if X is a totally bounded metric space and f is a continuous function from X onto a metric space Y, then Y must also be totally bounded? Of course not. We have seen continuous functions on totally bounded spaces such as 0, 1 whose ranges are not bounded. 10. Give an example of a metric space X and two closed subsets H and K of X such that H fact that inf d x, y x H and y K  0. K in spite of the Take X  R and Z H n n 2 and K n 1 n Z n 2 For another example, take X  R and 2 H x, 1 x x 1 K x, 1 x x 1 and 11. Suppose that X is a metric space, that H and K are closed subsets of X and are disjoint from each other and that the subspace K of X is compact. Prove that x H and y K  0. inf d x,...
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