$F(x,y) =x^2 y /x^2+y^2 , (0,0) $. The answer is that F is not continues at $(0,0)$ because
- $\lim (x,y)>(0,0) ~ f(x,y)$ does not exist
- $(0,0) \in D_f$
where $D_f$ is the domain of $f$.
I try a method where you let $y=g_1(x)=x$ And $y=g_2(x)=x^2$ which is it to curves that pass the point $(0,0)$ etc And the limit is equal which means that the function is exist !
And why does point $(0,0)$ belong to $D_f$ ?