Could you give me an example of the function $f(x)$ which is not Riemann integrable but is such that $xf(x)$ is Riemann integrable?

Question

I guess this is an easy exercise for analysts but at the moment i do not know of such an example.

So:

Could you give me an example of the function $f(x)$ which is not Riemann integrable but is such that $xf(x)$ is Riemann integrable?

Answer 1

Say $f(x)=0$, for $x=0$ and $f(x)=\frac{1}{x}$ for $x > 0$. Then $f$ isn't integrable on $[0,1]$ and but $xf(x)$ is.