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Analysis2001Aug - Analysis Preliminary Examination Fall...

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Unformatted text preview: Analysis Preliminary Examination Fall 2001 1. Let A be an uncountable set of real numbers. Prove that A has an accu- mulation point. 2. Let f (1:) be a differentiable mapping of the connected open subset V of IR". Suppose that f’ (z) = 0 on V, prove that f is constant on V. 3. Prove or disprove: the function f(2) = 23/210g1: is uniformly continuous on the interval (0,1). 4. Let f(z,y) = (11,11) where u = z“ - y2 and v = 22y describe a map from ill2 to R”. (a) What is the range of this map? (b) Show that if (u,v) ¢ (0,0) then f has an inverse in a. neighborhood of (u, v). (c) Show that there is no neighborhood of (0, 0) in which f has an inverse. 5. Prove that °° sin(n4x) 2 n2 n=1 defines a. continuous function on IR. . 6. (3) Find the limit 1 lim A e-Alvl dy. X400 _1 (b) Let g : R —+ R be a bounded, continuous function. For 1: E R, find the limit 1 lim A g(:: + y)e”\M dy. A-+oo __1 Hint: Try a “nice” 9 first, formulate a guess, and then try to prove your guess is correct. ...
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