Analysis2003Aug - Analysis Preliminary Exam If f is...

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Unformatted text preview: Analysis Preliminary Exam August 16 2003 . If f is continuous on [(1,1)] and Fm : /$f(t) dt for a: 6 [a,b], show that F’ : f on (a,b). . Prove that <i%)-lnn—>7 16:1 for some 7 E (1/2,1). .Letsz1—>R1 by f($)= { x, if :r, is irrational or :1: = 0 p sin %, if x : 2 Where p and q are integers with no common divisors. \Nhere is f continuous? . For each n let n : R1 —> R1 be a non-decreasin function and assume 'n conver res 7 E; point—Wise to a continuous function f. Prove that fn converges uniformly on compact sets to f. . Let f be a continuous function on [0,1] such that for all n 2 0.Show that f is identically zero. . Show that there is an open interval I containing 0 and a unique curve (:1:(t) y(t)),t E I with (a:(0),y(0)) : (1,1) satisfying (*) x+y2+sint=2 2:2 + ty2 2 1. Find the velocity of the curve at t = 0. For a given to E I is there a unique solution (33,31) to (*) with t = to? ...
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