Composition operators with closed range for one-dimensional smooth symbols

We give a sufficient condition for a smooth not necessarily injective function ψ:R→R ensuring that 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 which gave a description of composition operators Cψ with closed range for smooth injective symbols ψ:R→Rd.