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 December 26th, 2011, 01:52 PM #1 Senior Member   Joined: Nov 2011 Posts: 100 Thanks: 0 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 $f(x)=\begin{cases}-1,&x\in\mathbb{Q}\\1,=&x\in\mathbb{Q}^c\end{cases}=$. Then $f^2=1$ 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!
 December 27th, 2011, 06:42 AM #2 Senior Member   Joined: Nov 2011 Posts: 595 Thanks: 16 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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