[math-fun] algebraic geometry query

Given a plane curve C defined by f = 0 with f(x, y) polynomial, and a region R of the plane (possibly extending to infinity), I assert that C avoids R provided C avoids the boundary of R ; and C has no critical points within R . There must be a well-known theorem to this effect (unless, of course, it's actually false --- a situation by no means previously unknown). But I don't know a reference (or a counter-example) --- anybody? A straightforward way to locate the critical points seems to be to compute the discriminant g of f with respect to (say) x , then find the roots of g(y) = 0 . Is there a more respectable alternative? WFL