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Analysis2007Jan - and real continuous functions g i x on[0...

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Preliminary Exam Jan 2007 1. Let X be a metric space and let A j be subsets of X , j = 1 , 2 , . . . . For each of the following state- ments, prove it or give a counterexample (the means limit points): (i) ( A 1 A 2 ) A 1 A 2 (ii) j =1 A j j =1 A j 2. Prove that the series n =1 n 2 n ! is convergent and find its sum. 3. Let f : ( - 1 , 1) R be a di ff erentiable function such that f (0) = 0 and f (0) R exists. Prove that the limit lim x 0 f (2 x ) - 2 f ( x ) x 2 exists. 4. (a) Let f 4 R (this means f 4 is integrable dx on some closed interval) prove or disprove, f R . (b) Let f 5 R prove or disprove, f R . 5. Let f ( x, y ) be a real continuous function on the rectangle [0 , 1] × [0 , 2]. Given > 0 show that there exists n and real continuous functions
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Unformatted text preview: and real continuous functions g i ( x ) on [0 , 1] and h i ( y ) on [0 , 2] for i = 1 , . . . , n so that | f ( x, y )-n ² i =1 g i ( x ) h i ( y ) | < ± for all ( x, y ) in the rectangle. 6. Given the equations x-f ( u, v ) = 0 and y-g ( u, v ) = 0 (a) give conditions that assure you can solve for ( x, y ) in terms of ( u, v ) and (b) similarly that you can solve for ( u, v ) in terms of ( x, y ). (c) Assuming these conditions are satisfied prove that ∂x ( u, v ) ∂u ∂u ( x, y ) ∂x = ∂y ( u, v ) ∂v ∂v ( x, y ) ∂y 1...
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