4130prelim2sols - 4130 PRELIM 2: TAKE-HOME Due Tuesday...

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Due Tuesday April 20: no extensions. This is an exam. Unlike a normal homework, you are not allowed to work on these problems in groups. However, do not feel that you have to work in complete isolation: you may discuss the problems with the lecturer or the TA if you need help, or if you find the wording of the questions ambiguous. You are also free to use the textbook, lecture notes, and previous homeworks, but you are not supposed to use any other references. (1) Let A R . A function f : A R is said to satisfy a Lipschitz condition on A if there exists M R such that | f ( x ) - f ( y ) | ≤ M | x - y | for all x,y A . (a) Show that if f satisfies a Lipschitz condition, then f is uniformly continuous. Suppose f satisfies a Lipschitz condition on A . Then given ε > 0, take δ = ε/M . If | x - y | < δ then | f ( x ) - f ( y ) | < ε . So f is uniformly continuous on A . (b) Show that the function f ( x ) = p | x | with domain [ - 1 , 1] is uniformly continuous but does not satisfy a Lipschitz condition. The function f is uniformly continuous on [ - 1 , 1] because it is continuous, and [ - 1 , 1] is a compact set, so it is uniformly continuous. To see that f does not satisfy a Lipschitz condition, suppose for a contradiction that there is some M such that | p | x | - p | y || ≤ M | x - y | for all x,y [ - 1 , 1]. In particular, we can take y = 0 and x > 0, so x Mx for all x > 0. But then M 1 x for all x > 0, which is impossible. (2) [Extra credit.] Recall that for A,B subsets of R , we say that A is dense in B if A B and B is a subset of the closure of A . (a) Give an example of a set A such that A Q is not dense in A . 1
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4130prelim2sols - 4130 PRELIM 2: TAKE-HOME Due Tuesday...

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