Prove that $f(x)=x^2 \sin\left ( \frac{1}{x} \right )$ for $x \neq 0$, $f(0)=0$ is differentiable on $\mathbb{R}$.
Show that the function $g(x) = x^2 \sin(\frac{1}{x}) ,(g(0) = 0)$ is everywhere differentiable and that $g′(0) = 0$