Imaginary $\delta$ in proof of continuity

As it is mentioned in the comments $\delta$ might not be real.

If you want, you can do this:

Let $ \epsilon>0$

We have that $$|f(x,y)-f(x_0,y_0)|=|(x-x_0)+(y-y_0)| \leqslant$$ $$|x-x_0|+|y-y_0|= \sqrt{|x-x_0|^2}+ \sqrt{|y-y_0|^2} \leqslant$$ $$\sqrt{(x-x_0)^2+(y-y_0)^2}+ \sqrt{(x-x_0)^2+(y-y_0)^2}<2 \delta$$

Take $\delta= \epsilon /2 $ and you are done.