Proving that $f$ is differentiable at $0$

Question

Let's consider the following function:

$$f(x,y)=\begin{cases} (x^2+y^2)\sin\left(\dfrac{1}{x^2+y^2}\right) & \text{if }x^2+y^2\not=0 \\{}\\ 0 & \text{if }x=y=0 \end{cases}$$

I know that $f_x$ and $f_y$ are not continuous at $0$. How to prove that $f$ is differentiable at $0$?

Answer 1

Hint:

Change to polar coordinates and show

$$\lim_{r\to 0}\frac{r^2\sin(r^{-2})-0}{r}=0$$

Hint 2: The sine function is bounded.