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December 26th, 2011, 02:52 PM   #1
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Riemann integration

So I think I have part of the following problem, but am stuck on the other bit.

Suppose f is a bounded, real valued function on [a,b] and that f^2 is Riemann integrable on [a,b]. Does it follow that f is Riemann integrable? What if we assume that f^3 is Riemann integrable?

So I don't think that f^2 being Riemann integrable implies f is; consider the function . Then is certainly Riemann integrable on [a,b], but f is not (would need to show).

I'm not sure about if the hypothesis is changed to f^3 being Riemann integrable though. Does this imply that f is Riemann integrable? I think it might, but not sure..

Thanks!
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December 27th, 2011, 07:42 AM   #2
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Re: Riemann integration

Hi,

Yes I think your conterexample 1) is good to show that if f^2 is riemann integrable then it does not mean f is riemann integrable. For the other one I guess tath If f was complex then that would still be wrong since you can use the same kind of reasoning with the triple root of unity. if f has only real value (your problem), then I guess that using riemann definition of integration you should be able to derive your result using the fact that f^3 and f have the same sign everywhere.
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