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I have a question to ask please.

$f=x\sin(1/x)$ ($f(0)=0$);

$f=\sin x/x$ ($f(0)=0$);

$f=(1/x)\cos(1/x^2)$ ($f(0)=0$);

$x\in [0,1]$ for all the cases.

Are those functions integrable? That means that also they have a derivative?

Any help will be appreciated.

Nikos Ioannidis
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  • Check Lebesgue criteria for Riemann integrability. See here. – Mhenni Benghorbal Mar 17 '14 at 16:20
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    No, a function does not need to be continuous to be integrable, and yet differentiable functions are continuous. I suspect you may have instead wanted to ask whether functions that are integrable are necessarily derivatives. The answer is no anyway. One could also ask whether derivatives are necessarily integrable, and again the answer is no, see here. – Andrés E. Caicedo Mar 17 '14 at 16:30
  • Thanks for the responses but are those functions in my first post integrable?Do the have a derivative? – Nikos Ioannidis Mar 17 '14 at 16:36
  • For the specific functions you mention, review the limit definition of derivative, and see whether one can compute the relevant limits. – Andrés E. Caicedo Mar 17 '14 at 16:38
  • Just because a function is integrable, doesn't mean it's also differentiable (has a derivative). – Ben Grossmann Mar 17 '14 at 17:00

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