p13 - (14 points, Exercise 25(b) on p.102). Let be an...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #13 (due on Wednesday, December 7). 50 points are divided between 4 problems. #1. (10 points, Exercise 2 on p.98). If f is a continuous mapping of a metric space X into a metric space Y , prove that f ( E ) f ( E ) for every set E X . ( E denotes the closure of E ). Show, by an example, that f ( E ) can be a proper subset of f ( E ). #2. (12 points, Exercise 23 on p.101). Show that every increasing convex function of a convex function is convex. #3.
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Unformatted text preview: (14 points, Exercise 25(b) on p.102). Let be an irrational real number. Let C 1 be the set of all integers, let C 2 be the set of all n with n C 1 . Show that the algebraic sum C 1 + C 2 := { x = x 1 + x 2 : x 1 C 1 , x 2 C 2 } is not closed in R . #4. (14 points). Let f ( x ) be a function on R , which is continuous at each rational point q R . Show that f ( x ) is continuous at some irrational points....
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This document was uploaded on 02/15/2012.

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