How can I prove by definition the continuity of $f(x,y)=\frac{1}{\sqrt{x^2+y^2}}$ where $(x,y)\ne (0,0)$. How can I set the value at $(0,0)$ to make it continuous?
I think that the continuity in $(0,0)$ is not possible, using trajectories, but I'm not able to see which is the $\delta$ in the definition given any $\varepsilon>0$.