[math-fun] curves in regions

Lunnon: 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... WDS: a counterexample: C=unit circle, R = thickened version of unit circle (annulus).