3.II.9C

For a function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ state if the following implications are true or false. (No justification is required.)

(i) $f$ is differentiable $\Rightarrow f$ is continuous.

(ii) $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist $\Rightarrow f$ is continuous.

(iii) directional derivatives $\frac{\partial f}{\partial \mathbf{n}}$ exist for all unit vectors $\mathbf{n} \in \mathbb{R}^{2} \Rightarrow f$ is differentiable.

(iv) $f$ is differentiable $\Rightarrow \frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are continuous.

(v) all second order partial derivatives of $f$ exist $\Rightarrow \frac{\partial^{2} f}{\partial x \partial y}=\frac{\partial^{2} f}{\partial y \partial x}$.

Now let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be defined by

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

Show that $f$ is continuous at $(0,0)$ and find the partial derivatives $\frac{\partial f}{\partial x}(0, y)$ and $\frac{\partial f}{\partial y}(x, 0)$. Then show that $f$ is differentiable at $(0,0)$ and find its derivative. Investigate whether the second order partial derivatives $\frac{\partial^{2} f}{\partial x \partial y}(0,0)$ and $\frac{\partial^{2} f}{\partial y \partial x}(0,0)$ are the same. Are the second order partial derivatives of $f$ at $(0,0)$ continuous? Justify your answer.

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