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Unformatted text preview: Real Analysis - Math 630 Homework Set #4 - Chapter 4 by Bobby Rohde 9-28-00 Problem 1 & a) Show that if f & x & 0, x is irrational 1, x is rational Then R a b f & x x & b a and R a b f & x x & & Proof R & &&& a b f x x inf & a b x x , step functions ( x ) f ( x ), x . But each step function is composed of a finite collection of open intervals, and in the domain each open interal an rational number since the rationals are dense. Thus ( x ) f ( x ), x implies that the value over any open interval of the step function ( x ) is 1. Thus the infimum of the integral of all such step functions must be when ( x ) = 1, x . R & &&& a b f x x inf & a b x x b a 1 b a . R & &&& a b f x x sup & a b x x , step functions ( x ) f ( x ), x . Since irrationals are dense, each interval in the composition of has an irrational in its domain. Thus has an irrational in its domain....
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- Spring '08