Characterization of composition operators with closed range for one-dimensional smooth symbols

We give a full characterization of smooth symbols ψ:R→R for which the composition operator Cψ:C∞(R)→C∞(R), F↦F∘ψ has closed range. This generalizes in a special case the result of Kenessey and Wengenroth who gave such a characterization for smooth injective symbols ψ:R→Rd.