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**arbolis** Let $\displaystyle f(x,y)= \begin{cases} \frac{x|y|}{\sqrt{x^2+y^2}} \hspace{2cm} \text{if (x,y)} \neq (0,0) \\ 0 \hspace{3.1cm} \text{if (x,y) =0} \end{cases}$.

Demonstrate that $\displaystyle f$ has directional derivatives in all directions in $\displaystyle (0,0)$ but that $\displaystyle f$ is not differentiable in $\displaystyle (0,0)$.

My attempt :

I notice I can't use the theorem that states that if $\displaystyle f$ is differentiable in a point, then it has directional derivatives in all directions in this point.

I believe $\displaystyle f$ is continuous in $\displaystyle (0,0)$... I didn't prove it via the definition but I've failed at showing that it is not continuous. I realize that it would have been easy if $\displaystyle f$ wasn't continuous because $\displaystyle f$ could not be differentiable but this is not the case.

I'm stuck at starting the exercise.

I guess I have to use the definition to show that $\displaystyle f$ is not differentiable, i.e. that there does not exist a linear function $\displaystyle \varphi$ such that $\displaystyle \lim _{\vec h \to 0} \frac{f(\vec {x_0} + \vec h)-f(\vec x_0)- \varphi (\vec h)}{\| h \|}=0$.

Seems quite hard, I don't know how to start.