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- Apr 14, 2013

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We consider the function\begin{align*}f:\mathbb{R}^2 &\rightarrow\mathbb{R} \\ (x,y)&\mapsto \begin{cases}\frac{x^3}{x^2+y^2} & \text{ if } (x,y)\neq(0,0) \\ 0 & \text{ if } (x,y)=(0,0) \end{cases}\end{align*}

(a) Show that all directional derivatives of $f$ in $(0,0)$ exist.

(b) Is $f$ differentiable in $(0,0)$ ?

For (a) Ihave donethe following:

$$\partial_vf(0,0)=\lim_{h\rightarrow 0}\frac{f(hv_1, hv_2)-f(0,0)}{h}=\lim_{h\rightarrow 0}\frac{v_1^3}{v_1^2+v_2^2}=\frac{v_1^3}{v_1^2+v_2^2}\in \mathbb{R}$$ The limit exists for all $(v_1,v_2)\neq (0,0)$ and so all directional derivatives of $f$ in $(0,0)$ exist.

Is that correct?

For (b) I thought to find a sequence $a_n$ that converges to $(0,0)$ such that $f(a_n)$ doesn't converge to $0$, but I haven't found yet such an examle. Can we maybe not find such an example because $f$ is differentiable in $(0,0)$ ?