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Unformatted text preview: on [0 , ) but not directly Riemann integrable. 4. Suppose f : [0 , ) [0 , ) is such that for any sequence a n of nonnegative terms we have X n =1 a n < = X n =1 f ( a n ) < Prove that lim sup x + f ( x ) x < 5. Let f be a continuous real valued function dened on the unit square and for each x 1 let f x be the function on the unit interval dened by f x ( y ) = f ( x, y ) . Prove that for any sequence x n in [0,1] there is a subsequence n k such that f x n k converges uniformly on [0,1]. 6. If c is a real parameter prove that x 7 + x + c = 0 has a unique real root and that this root is a dierentiable function of c ....
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This note was uploaded on 06/19/2011 for the course MATH 600 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.
- Spring '11