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# hw5 - f is continuous then at least one of the functions f...

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Math 125A, Fall 2009 Homework 5 Due Date: October 30, 2009 Problem 1: De ne a sequence of functions f n : R ! R by: f n ( x ) = ( 1 ; if n x n + 1 0 ; otherwise (a) Prove that f n ! 0 pointwise. (b) Prove, however, that f n does not converge to 0 uniformly. Problem 2: De ne a sequence of functions f n : R ! R by: f n ( x ) = x 2 n 1 + x 2 n (a) Determine the pointwise limit f of this sequence. (b) Is the convergence to f uniform? Justify your answer. Problem 3: Determine whether the statement is true or false. If the statement is true, then prove it. Otherwise, provide a counterexample. (a) If f n ! f pointwise, and if each f n is continuous, then f is continuous also. (b) If f n ! f pointwise, and if R 1 0 f n ( x ) dx = 1 for all n , then R 1 0 f ( x ) dx = 1 also. (c) If f n ! f uniformly, and if
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Unformatted text preview: f is continuous, then at least one of the functions f n must be continuous. Problem 4: Let ( f n ) be a sequence of monotonically increasing functions which converges pointwise to f . Prove that f must also be monotonically increasing. Problem 5: Let f n : [ a;b ] ! R be a sequence of continuous functions, and ( x n ) be a sequence in [ a;b ] converging to x . If f n ! f uniformly, then prove that f n ( x n ) ! f ( x ). Problem 6: De±ne a sequence of functions f n : R ! R by: f n ( x ) = n X k =1 1 x 2 + k 2 Prove that f n converges uniformly on R ....
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