Continuity of $f(x,y)$ which is not defined along a path

Question

Consider $f(x,y) = x^2+y^2$ when $x-y \neq 0$ and $0$ if $(x,y) = (0,0)$.

This function is easily shown to be continuous along all paths, but along $x=y$ it is not defined!

So will it said to be continuous at $0,0$ ?

Answer 1

For the continuity at $(0,0)$ it doesn't mind if $f(x,y)$ is not defined for $x=y\neq 0$, indeed we have that trivially

$$\lim_{\substack{(x,y)\to(0,0)}\\\quad \:x\neq y}x^2+y^2=0=f(0,0)$$

therefore by definition $f(x,y)$ is continuous at $(0,0)$.