Homework_set_6_typeset - Math 413, Fall semester, 2007...

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Math 413, Fall semester, 2007 Bigtimes Homework set 6, handed out September 27, due October 4 Remember that there is a prelim scheduled for October 4, 7:30 pm. I don’t know the room yet, but I will announce it as soon as I do. The curriculum is the text through Chapter 4. You should have read everything through Chapter 4. Here are some review exercises. Problem 1. Consider the function f : R R defined by f ( x )= 1 q if x Q and x = p q with p, q coprime 0i f x is irrational. Show that f is continuous at all irrationals. Problem 2. A function f :[ a, b ]is convex if for all x, y satisfyling a x<y b and all t (0 , 1) we have f ( tx +(1 - t ) y ) tf ( x )+(1 - t ) f ( y ) . (a) Show that every convex function is continuous. (b) Show that if f is convex, then so is e f . Show that f a, b ] R is convex if and only if it satisfies
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