Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.
You've proved continuity and that $f(0) = 0$. To prove differentiability, you need to prove that $$\lim_{x\to 0} \frac{f(x) - f(0)}{x-0}$$ exists. However, this simplifies to proving $$\lim_{x\to 0} \frac{f(x)}{x}$$ exists. And by the given condition, we see $$\left | \frac{f(x)}{x} \right| \le \lvert x \rvert .$$ Thus $$\lim_{x\to 0} \frac{f(x)}{x} = 0$$ which shows that $f$ is differentiable at $0$ with $f'(0)=0$.