My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Reply
 
LinkBack Thread Tools Display Modes
April 30th, 2010, 11:10 PM   #1
Newbie
 
Joined: Apr 2010

Posts: 2
Thanks: 0

Riemann/Darboux Integral

Hello to everyone.
I have a question. Is the Riemann Integral really equivalent to Darboux Integral?.
For example, consider the function f(x)=sin(x)/x. I want to know if it is integrable between -1 and 1. The problem arises at x=0.
Now, this function is clearly Darboux integrable, since you solve the problem of the non-existance of f(0) with the supremum operator. But I don't think it is Riemann integrable, since you can not evaluate f(0), and the Riemann definition clearly states that the sum must exist for ANY election of evaluation points. If you defined f(0) as 1, you would not be solving the problem, because you would be integrating a different function,not f(x), since you are changing the domain of the function.
So, is it possible to say that the Riemann and Darboux integrals are equivalent iff the function is well defined for all points in the interval?.
Please correct me if I am wrong.
I will deeply appreciate any replys.
DavidDante is offline  
 
May 1st, 2010, 12:46 PM   #2
Global Moderator
 
Joined: May 2007

Posts: 6,612
Thanks: 617

Re: Riemann/Darboux Integral

I would call your question a quibble, since you want to leave sinx/x undefined at x=0, rather than accepting any value at all. Whatever you define it to be, the resultant function will be Riemann integrable, since the x interval containing x=0 can be made as small as you want, so its contribution -> 0, no matter what the (finite) value assigned at that point.
mathman is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
integral, riemann or darboux



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
riemann integral!!! sedasen Calculus 1 March 25th, 2013 08:25 AM
Riemann sum/Integral ralphbs Calculus 2 January 22nd, 2013 07:56 PM
Riemann/Darboux Integral DavidDante Real Analysis 2 May 1st, 2010 08:32 PM
Riemann vs. Riemann-Stieltjes integral cheloniophile Real Analysis 1 November 23rd, 2008 05:30 PM
riemann integral access_nash Calculus 1 November 9th, 2008 10:29 PM





Copyright © 2018 My Math Forum. All rights reserved.