Usually when we have a monotonic function $f(x)$ and we know it has a root in $[a,b]$, we can repetitively apply binary search to the region $[a,b]$ so that you get an root $\epsilon$ close to the real root for any arbitrary $\epsilon$.
So naturally my questions asks about a function such as $f(x,y)$. What conditions are needed to perform binary search? If I know that it has a solution in the circle $x^2+y^2 \leq r^2$, can I somehow approximate $(x,y)$ such that if the real roots are $(x*, y*)$ then the binary search returns $(x',y')$ where $\sqrt{(x'-x*)^2+(y'-y*)^2} \leq \epsilon $ ?