True or false:
Let $f:(0,1)\to \Bbb R$ be a twice differentiable function, then, if $f$ and $f'$ bounded, then $f''$ bounded.
I tried integrating $x\sin (1/x)$, but $F(x)=\int x\sin (1/x) dx$ seems to be unbounded.
Does it have to do something with $f'$ bounded then $f$ uniformly continuous?
How about $x^{3/2}$? First derivative is $\frac{3}{2} x^{1/2}$, second is $ \frac{3}{4} x^{-1/2} $.