Exercise 22 solution: Let 0 < ϵ ≤ 1. Let x ∈ R. F_{ω,ϵ}(x + 1) = x + 1 + ω + \frac{ϵ}{2π} sin(2π(x + 1)) = F_{ω,ϵ}(x) + 1. Hence F_{ω,ϵ} is the lift of a circle map f_{ω,ϵ}.

Since ϵ ≤ 1, F′_{ω,ϵ}(x) = 1+ϵ cos(2πx) = 0 if and only if ϵ = 1 and x = 0.5+k, k ∈ Z. If ϵ < 1 then F′_{ω,ϵ}(x) > 0 for all x and F_{ω,ϵ} is a diffeomorphism. If ϵ = 1, the reader may check that x = 0.5 + k is an inflection point of F_{ω,1}. If x ≠ 0.5 + k then F′_{ω,1}(x) > 0, so that F_{ω,1} is strictly increasing and thus a homeomorphism. We have F_{ω,ϵ} is the lift of an orientation preserving circle homeomorphism f_{ω,ϵ} for 0 < ϵ ≤ 1.