Conjugations between circle maps with a single break point

Consider two circle homeomorphisms fi∈C2+α(S∖{bi}), α>0, i=1,2 with a single break point bi i.e. a discontinuity in the derivative Dfi, and identical irrational rotation number ρ. Suppose the jump ratios and do not coincide. Then the map ψ conjugating f1 and f2 is a singular function i.e. it is continuous on S1 and Dψ(x)=0 a.e. with respect to Lebesgue measure.