ps10 - Part 2 3 Problem 20 from page 169 4 Problem 24 from...

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18.100B and 18.100C Fall 2011 Problem Set 10 Due December 8th at 4 pm in room 2-108. Hand in parts 1, 2 and 3 separately. Put your name and whether you are registered for 18.100B or 18.100C on each part. Part 1 1. Let K : [0 , 1] × [0 , 1] R be continuous, and let F be the family of functions f from [0 , 1] to R satisfying f ( x ) = Z 1 0 K ( x,y ) g ( y ) dy for some continuous function g : [0 , 1] [ - 1 , 1]. Prove that the family F is equicontinuous. 2. Problem 16 from page 168.
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Unformatted text preview: Part 2 3. Problem 20 from page 169. 4. Problem 24 from page 170. Part 3 5. Problem 1 from page 196. 6. Let f : [0 , 1] × [0 , 1] → R be continuous, and suppose g y ( x ) = f ( x,y ) is continuously differentiable with respect to x , and g y ( x ) is continuous with respect to y . Prove that d dx Z 1 f ( x,y ) dy = Z 1 g y ( x ) dy. 1...
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