Conjugations between circle maps with a single break point

Consider two circle homeomorphisms f i ∈ C 2 + α ( S ∖ { b i } ) , α > 0 , i = 1 , 2 with a single break point b i i.e. a discontinuity in the derivative Df i , and identical irrational rotation number ρ. Suppose the jump ratios σ 1 = Df 1 ( b 1 − 0 ) Df 1 ( b 1 + 0 ) and σ 2 = Df 2 ( b 2 − 0 ) Df 2 ( b 2 + 0 ) do not coincide. Then the map ψ conjugating f 1 and f 2 is a singular function i.e. it is continuous on S 1 and D ψ ( x ) = 0 a.e. with respect to Lebesgue measure.