Say $F$ is a complicated real function of two variables, of which we know just a defining property, but we definitely don't have continuity, and for what it matters the defining property might even end up being contradictory - still, even if it's in order to show it's not the case, we may assume this $f$ is well defined.
If the defining property implyes that $F(x,y)=R(x,y)$ where $R$ is a rational function, is it safe to take limits on both sides, deducing information about $F$? What is the theorem that allows/forbids it?