I have this function
$$f(x)=\begin{cases}x^a \cos\big(\frac{1}{x}\big)&x>0\\0&x=0\end{cases}$$
defined on $[0,1]$. I found that $f$ is bounded, differentiable and continuous and for the last part I need to prove that $f$ is Riemann integrable on $[0,1]$. I got a bit stuck (continuity and boundedness imply Riemann integrability, but this doesn't work the other way around, so I can't require $f$ to be continuous or bounded necessary). Can anyone help me?