Let f(x)=x^2sin 1/x for x≠0 and f(0)=0.

(A) true: $\dfrac{f(h)-f(0)}{h}=h\sin\dfrac1h$ and $|h\sin(1/h)|\le|h|\to0$, so $f'(0)=0$. For $x\ne0$, $f'(x)=2x\sin\dfrac1x-\cos\dfrac1x$. (B) false: as $x\to0$ the first term tends to $0$ but $\cos(1/x)$ oscillates between $-1$ and $1$ without a limit, so $f'(x)\not\to f'(0)=0$. (C) true: $|f'(x)|\le2|x|+1<3$ on $(-1,1)\setminus\{0\}$, and $f'(0)=0$ as well. (D) false: $\dfrac{f'(h)-f'(0)}{h}=2\sin\dfrac1h-\dfrac{\cos(1/h)}{h}$ is unbounded and has no limit as $h\to0$. Thus $f$ is differentiable everywhere with a bounded but discontinuous derivative, so exactly (A) and (C) hold.