Did you think about using some equivalent metrics? The taxicab metric seems to do the trick.
If you're unfamiliar, it is
.
Good luck.
I want to check, using the epsilon-delta-definition of continuity, that the function f(x,y)=x+y is continuous. But I can't seem to find a way to show that for epsilon > 0, sqrt((x-x')^2+(y-y')^2) < delta implies |x + y - x' - y'| < epsilon. Any help?