Yeah, I know that.
I don't have a clue how to show that, any idea?
It is not true.
Different paths give different answers.
The path $\displaystyle x=y$ gives the limit $\displaystyle 1$.
While the path $\displaystyle x=0$ gives the limit $\displaystyle 0$.
Thus, the limit does not exist.
Thus, the function is not continous.