1873_solutions

X a challenge question would be to ask whether the

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Unformatted text preview: a, it would follow from the fact that g  fg f that g is continuous at a. Therefore f  g cannot be continuous at a. b. What can we say about the continuity of the function fg at the point a? If f a 0 then, since the equation fxgx gx  fx would hold for every number x sufficiently close to a, we could use an argument like the one we used in Part a to deduce that the function fg cannot be continuous at a. However, execise 2 shows us to functions, one continuous at 0 and the other discontinuous at 0 whose product is continuous at 0. c. What can we say about the continuity of the function fg at the point a if f a  0 and g is a bounded function? Choose a number p such that |g x |  p for each x S. Whenever x S we have |f x g x f a g a |  |f x g x | p|f x | and we can now use this inequality to prove that fg must be continuous at a. Suppose that  0. Using the fact that f is continuous at w and the fact that /p  0 we choose a number   0 such that the condition |f x |  p whenever x S a , a   . Then, whenever x B a,  we have |f x g x faga | p|f x |  p p . d. What can be said about the continuity of the function fg if f a 0? The purpose of this part of the question is to prod people who may not have done Part a. 16. Give an example of two functions f and g that are both discontinuous at a given point a such that their sum f  g is continuous at a. We define 1 if x  2 fx  0 if x  2 1 1 if x  2 and gx  if x  2 0 if x  2 1 if x  2 17. a. Given that f is a continuous function from a closed subset H of R into R and that a set 223 H, prove that the E x H fx fa is closed. Solution: Suppose that x n is any convergent sequence in the set E. We shall show that the limit of this sequence, that we shall call x, must also belong to E. Since the set H is closed we know that x H and therefore we know that f is continuous at the number x. Therefore lim f x  n Ý f xn  n Ý f a  f a lim and so x E as we promised. b. Given that f and g are continuous functions from a metric space X to a metric space Y and that E x X fx gx prove that the set E must be closed in X. This exercise is very similar to part a. Suppose that x n is any convergent sequence in the set E. We shall show that the limit of this sequence, that we shall call x, must also belong to E. We now observe that lim f x  n Ý f xn  n Ý g xn  g x lim and so x E as we promised. 18. a. Given that f and g are continuous functions from R to R and that f x  g x for every rational number x, prove that f  g. The set of numbers x for which f x  g x is closed and includes the set Q of rational numbers. Therefore this set is all of R. b. Given that f : Z  R, that f 1  1 and that the equation f xt  f x f t holds for all positive integers x and t, prove that f x  x for every positive integer x. Solution: If you are familiar with the method of proof by mathematical induction which is available in this book in an optional section then you should use mathematical induction to do this exercis...
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