Prove that function is not continuous in $(0, 0)$

Question

We have function $f:\mathbb R^2 \to \mathbb R$, $$ f(x,y) = \begin{cases} \frac{xy}{x+y}, & \text{if $x+y \neq 0$} \\[2ex] 0, & \text{if $x+y=0$} \end{cases} $$ And the problem asks to demonstrate that $f$ is not continuous in $(0,0)$, and to demonstrate that it admits derivatives after any direction from $(0, 0)$.

Answer 1

$f (\frac 1 n, -\frac1 {n+1})=-1$ for all $n$ and $f(0,0)=0$.

Answer 2

Rewrite the expression as $$\frac 1{\dfrac1x+\dfrac1y}.$$

Then with

$$\dfrac1y=\frac1a-\dfrac1x$$

$x$ and $y$ tend to zero simultaneously, while the limit is $a$.