This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Real Analysis  Math 630 Homework Set #4  Chapter 4 by Bobby Rohde 92800 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....
View Full
Document
 Spring '08
 Staff
 Math

Click to edit the document details