$\newcommand\R{\mathbb R}$Suppose $f:\R^2 \to \R^2$ is a Whitney map with singularities (well, I'm not sure if this is the name for it, Whitney calls them excellent maps in his 1955 paper), i.e. it is an infinitely differentiable map such that the singularity set in the domain is a planar curve (possibly many of them) that is either smooth or have singularities that are cusps (i.e. the singularities are stable).

Then is it true that if we restrict $f$ to a connected component of the complement of the preimage of the critical values of $f$, $f$ is injective? This feels natural because according to Whitney $f$ is equivalent to a projection of a connected surface to the plane (I don't know how this is proven though, is there anywhere I can see the proof of this? I'm not sure if I see this in Whitney's paper). And I can imagine that if I remove the preimage of the critical values from this surface then I get homeomorphisms when restricted to the connected components. But I don't know how one proves this cleanly and rigorously. Any ideas or counterexample if this is not true? If it makes it any easier, what if we assume coordinates of $f$ are polynomials (so just a morphism between affine planes as varieties if you like).