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Unformatted text preview: X . Problem 6. Let f : ( a,b ) R be continuous. Prove that given x 1 ,...,x n in ( a,b ) that there exists an x ( a,b ) such that f ( x ) = 1 n ( f ( x 1 ) + + f ( x n )) . Problem 7. Let f : ( a,b ) R . Given c ( a,b ), show that f is dierentiable at c if and only if there exists a constant M so that f ( x ) = f ( c ) + M ( x-c ) + r ( x ) where r ( x ) satises lim x c r ( x ) x-c = 0 . Problem 8. Suppose that the series n =1 f n ( x ) converges uniformly on A and that g : A R is bounded. (a) Prove that the series n =1 g ( x ) f n ( x ) converges uniformly on A . (b) Show by example that the boundedness of g is necessary for part (a). 1...
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This note was uploaded on 03/02/2010 for the course MATH 28 taught by Professor Staff during the Spring '08 term at UMass (Amherst).
- Spring '08