All topology pairs $(X,Y)$ such that $f: X \to Y$ is continuous.

Question

Given an arbitrary function, or more specifically if you want let $R$ be a ring and let $X = S \times S; Y = R; S \subset R$ and $f(a,b) = a - b$, is there something interesting about all the topology pairs $(X,\tau_1), (Y, \tau_2)$ such that $f : X \to Y$ is continuous?