final - Intro to Modern Analysis Section II Summer MMX...

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Unformatted text preview: Intro to Modern Analysis Section II Summer MMX Final Exam Name DIES IOVIS AD III KAL AVG MMDCCLXIII AUC * Do all problems, in any order. Explain all answers. An unju ified response alone may not receive full credit. No notes, texts, or calculators may be used on this exam. Problem Possible Points Points Earned 1 10 2 10 3 10 4 10 5 10 6 10 TOTAL 60 GOODLUCK * August 12, 2010 1. (a) Suppose that f , f is continuous on [ a,b ] and that R b a f dx = 0 . Prove that f ( x ) = 0 for all x [ a,b ] . (b) Suppose f R ([ a,b ]) is not continuous and that R b a f dx = 0 . Does it follow that f ( x ) = 0 for all x [ a,b ] ? Explain. 2. Let ( X,d ) be a connected 1 metric space and f : X Z be a function such that for all i Z , f- 1 ( i ) is open in X . Prove that f is constant. 3. Suppose < a < 1 and b R . Fix x R and recursively define a sequence: for n > , set x n = ax n- 1 + b ....
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This note was uploaded on 10/18/2011 for the course MATH S4061X taught by Professor Peters during the Spring '11 term at Columbia.

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final - Intro to Modern Analysis Section II Summer MMX...

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