R-continuity of step function

Question

Let us define a function $f\colon\mathbb R^2\to\mathbb R$ by

$$ f(x,y) = \begin{cases} 1 & \text{if $xy\le 0$,} \\ 0 & \text{if $xy>0$.} \end{cases} $$

Does limit of $f$ as $(x,y)$ tends to $(0,0)$ exist?

Answer 1

No. In every neighborhood of $(0,0)$ you can find points $(x,y)$ with $xy>0$ as well as points with $xy\le 0$. Hence $f$ takes values $0$ and $1$ in every neighborhood of $(0,0)$ and is not continuous by the $\varepsilon$-$\delta$-definition of continuity.

Answer 2

For $t>0$ we have $f(t,t)=0$ and $f(t,-t)=1$. Conclusion ?