Non continuous function

Question

I'm struggling to answer this question : Show that this function admits partial derivatives in every direction without being continuous in $(0,0)$:

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

I have no problem showing that f admits partial derivatives in every directions, but I'm struggling to show that it is not continuous.

Answer 1

Hint : What is $f\left( \frac{1}{n}, e^{-n^2}\right)$ ? What should be the limit of this sequence if $f$ was continuous ?

Answer 2

Hint: Given a constant $c$, can you find a curve along which $(x,y)\to(0,0)$ and $f(x,y) \equiv c$?