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April 30th, 2010, 11:21 PM   #1
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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 fon ANY election of evaluation numbers. 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 right if i say that the Darboux and Riemann integral are equivalent iff f(x) is well-defined for all the points in the interval?
I will deeply appreciate any replies.
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May 1st, 2010, 12:52 PM   #2
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Re: Riemann/Darboux Integral

Further comment: It is no more Darboux integrable than Riemann integrable if it is undefined at x=0.
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May 1st, 2010, 08:32 PM   #3
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Re: Riemann/Darboux Integral

Thank you for your answer man.
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