Explain what it means for a function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{1}$ to be differentiable at a point $(a, b)$. Show that if the partial derivatives $\partial f / \partial x$ and $\partial f / \partial y$ exist in a neighbourhood of $(a, b)$ and are continuous at $(a, b)$ then $f$ is differentiable at $(a, b)$.

Let

$$f(x, y)=\frac{x y}{x^{2}+y^{2}} \quad((x, y) \neq(0,0))$$

and $f(0,0)=0$. Do the partial derivatives of $f$ exist at $(0,0) ?$ Is $f$ differentiable at $(0,0) ?$ Justify your answers.