Composition operators with closed range for smooth injective symbols R→Rd

In 1998, Allan, Kakiko, OʼFarrell, and Watson proved a description of the closure (with respect to the uniform convergence of all derivatives on compact sets) of A(ψ)={F∘ψ:F∈E(Rd)} for a smooth injective symbol ψ:R→Rd in terms of formal Taylor series. In that article it was conjectured that A(ψ) is closed if ψ is proper and has only critical points of finite order. In the present paper we first give a simple counterexample and then rectify the conjecture by adding a geometrical property for the curve ψ(R). This yields a characterization of .