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Analysis2010aug - • f is not Riemann integrable on[0 1...

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August 2010 Preliminary Exam in Analysis 1. Suppose that f : R R is a function such that f ( f ( x )) = x for all x R . Prove that there exists an irrational number t such that f ( t ) is also irrational. 2. Find three subsets A, B, C of the real line R such that A B = A C = B C = and A = B = C = R . Prove that your sets satisfy these properties. 3. Let X and Y be metric spaces. Suppose that f : X Y has the following property: for any continuous function g : Y R the composition g f is a continuous function from X to R . Prove that f is continuous. 4. Suppose that f : R R is a function such that f 0 ( x ) exists for all x R and f 0 ( - x ) = - f 0 ( x ) for all x R . Prove that f ( - x ) = f ( x ) for all x R . 5. Give an example of a bounded function f : [0 , 1] R
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Unformatted text preview: • f is not Riemann integrable on [0 , 1] • The function g defined by g ( x ) = sin f ( x ) is Riemann integrable on [0 , 1] Prove your claims using the definition of the Riemann integral. 6. Let f : R 3 → R 3 be a mapping defined by y 1 = x 1 + x 2 y 2 = x 2-x 1 y 3 = x 5 3 (a) Determine all points a ∈ R 3 at which f satisfies the assumptions of the Inverse Func-tion Theorem. (b) Is f an open mapping? Prove or disprove. Reminder. A mapping f : R 3 → R 3 is open if f ( W ) is an open subset of R 3 for every open set W ⊂ R 3 ....
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