Hw5 - Tufts University Department of Mathematics Math 136 Homework 5 This assignment is due Tuesday in class 1(15 points Let > 0 Let f Rn R be

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Tufts University Department of Mathematics Math 136 Homework 5 This assignment is due Tuesday, March 11, 2008 in class. 1. (15 points) Let ± > 0. Let f : R n R be bounded and let R be a nonempty subset of R n . Assume that for all x 1 R and all x 2 R , | f ( x 1 ) - f ( x 2 ) | < ± . Prove that sup R f - inf R f ± . This will prove Theorem 3 on the hints to Homework 4. 2. (25 points) Let A R n be bounded and let f : A R and g : A R both be bounded and integrable. As usual, let f and g denote the extensions to R . Let B be a rectangle such that A B . (a) Let R be a nonempty subset of B . Prove inf R f + inf R g inf R ( f + g ) sup R ( f + g ) sup R f + sup R g . (b) Let ± > 0. Prove there is a partition P of B such that both of the following hold for this same partition Z f - ± 2 < L ( f,P ) U ( f,P ) < Z f + ± 2 Z g - ± 2 < L ( g,P ) U ( g,P ) < Z g + ± 2 (c) Use the result of parts (a) and (b) to prove that f + g is integrable and R A ( f + g ) = R A f + R
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This note was uploaded on 03/27/2008 for the course MATH 136 taught by Professor Quinto during the Spring '08 term at Tufts.

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