4130prelim2

# 4130prelim2 - 4130 PRELIM 2 TAKE-HOME Due Tuesday April 20...

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4130 PRELIM 2: TAKE-HOME 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. (b) Show that the function f ( x ) = p | x | with domain [ - 1 , 1] is uniformly continuous but does not satisfy a Lipschitz condition. (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

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## This note was uploaded on 09/22/2010 for the course MATH 413 at Cornell.

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4130prelim2 - 4130 PRELIM 2 TAKE-HOME Due Tuesday April 20...

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