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

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Intro to Modern Analysis Section II Summer MMX Final Exam Name DIES IOVIS AD III KAL AVG MMDCCLXIII AUC * GOODLUCK * August 12, 2010
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1. (a) Suppose that f 0 , 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 0 < a < 1 and b R . Fix x 0 R and recursively define a sequence: for n > 0 , set x n = ax n - 1 + b . (a) Prove that { x n } n N converges. What is its limit? (b) Suppose f : R R is a continuously differentiable function such that f f ( x ) = ax + b . Show that f 0 ( f ( x )) f 0 ( x ) = a for all x R . (c) Show that f 0 ( x ) = f 0 ( ax + b ) for all x R . (d) Use parts (a) and (c) to show that f ( x ) = Ax + B for some A, B R . What are A and B ? 4. Suppose that { a n } n N is a decreasing sequence of non-negative reals. (a) Show that the series a n converges if and only if the series (2 n + 1) a n 2 converges. (b) Does the series n =0 e - n converge or diverge? 5. (a) Show that the continuous function f ( x ) = 1 /x is not uniformly continuous on
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