Unformatted text preview: MATH 410: Homework 6 Section 3.6 Due in class Friday 3/9/2012 1. Define f : [0, ) R by f (x) = 1 + x2 . First prove that f fits the hypotheses for Theorem 3.29 in the book and then elaborate on what the consequence of this theorem is. Note: This is not a book problem. 2. Let D = [0, 1] (2, 3] and define f : D R by f (x) = x if 0 x 1 x  1 if 2 < x 3 ** * Prove that f is continuous. Determine f 1 and prove that f 1 is not continuous. Does this contradict Theorem 3.29 in the book? Note: This is book problem 5. Section 3.7 1. Prove using sequences that: (a) lim (b)
x1 x4 1 x1 = 4 1 lim x1 = 2 x1 x1 * * Note: This is book problem 2. 2. Suppose that f : R R has the property that there is some M > 0 such that f (x) M x2 for all x. Prove that f (x) =0 lim f (x) = 0 and lim x0 x0 x Note: This is book problem 9. ** ...
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 Spring '08
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 Math, Topology, Book problem

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