The following function is not defined for $x=0$ and $y=0:$ $$f(x,y) = \frac {xy} {x^2+y^2}$$ Is it possible to add there a function value in such way that the modified function is continuous at zero point?
I tried to find the limit $\lim\limits_{(x,y)\to (0,0)} f(x,y)$.
If $\ y=0,$ then $f(x, 0) = \frac 0{x^2} = 0$,
if $\ x=0,$ then $f(0, y) = \frac 0{y^2} = 0$.
But for example:
If $\ y=x,$ then for all $x \neq 0$ is $f(x, x) = \frac{x^2} {x^2+x^2} = \frac 12$
Since I have obtained different limits along different paths, the given limit does not exist and the function is discontinuous.
I am not sure if I am right, but I think that is not possible to add a function value in such way that the modified function is continuous at zero point.
Any help is appreciated.
Thanks in advance.