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 diﬀerentiable at c if and only if there exists a constant M so that f ( x ) = f ( c ) + M ( xc ) + r ( x ) where r ( x ) satisﬁes lim x → c r ( x ) xc = 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...
View
Full Document
 Spring '08
 STAFF
 Math, Calculus, Topology, Metric space, Compact space

Click to edit the document details