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Analysis1992Aug - GRABUATE PRELIMINARYEX’AMINATION'”...

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Unformatted text preview: GRABUATE PRELIMINARYEX’AMINATION'” Analysis (Fall 1992) . Let {Ian} be a sequence of complex numbers converging to a. Show that n , 1 11m — 2 xj = a. n—ooo n _ J=1 , a) If fn 6 01(0, 2), n = 1,2,..., and f,’, converges uniformly to zero, while fn(1) converges to 1, prove that fn converges uniformly on (0,2). b) Is the result true if each fn is only differentiable on (0, 2)?‘ . Let (X, p) be a compact metric space and (Y, (1) be a metric space. a) If f : X ——> Y is continuous and onto show that (Y, d) is complete. b) If f is also one—to—one prove that f ‘1 : Y —> X is continuous. . Suppose f : R2 -—> R is 01. If fry exists in a neighborhood of (0,0) and is continuous at (0,0), prove that fyz exists at (0,0) and fyz (0, 0) = fly (0,0). . Let p(:z:,y) = (my — 1)2.+ 1:2 for (:c,y) 6 R2. Find inf{p(:r,y) : (gay) 6 R2}. . Suppose f is continuous and greater than 1 on [0, 1]. Prove that for positive a gag/Dime) Iadz)%=exp(/Ollnlf<x>_ldz) Hints: First establish the limit formally. Then attend to the intermediate results that require justification. ...
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