Let $f: \mathbf{R}^{2} \rightarrow \mathbf{R}$ be a function. What does it mean to say that $f$ is differentiable at a point $(a, b)$ in $\mathbf{R}^{2}$ ? Show that if $f$ is differentiable at $(a, b)$, then $f$ is continuous at $(a, b)$.

For each of the following functions, determine whether or not it is differentiable at $(0,0)$. Justify your answers.

(i)

$$f(x, y)= \begin{cases}x^{2} y^{2}\left(x^{2}+y^{2}\right)^{-1} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$$

(ii)

$$f(x, y)= \begin{cases}x^{2}\left(x^{2}+y^{2}\right)^{-1} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$$