Let fi∈C2+α(S1∖{ai,bi}), α>0,i=1,2, be circle homeomorphisms with two break points ai,bi, that is, discontinuities in the derivative Dfi, with identical irrational rotation number ρ and μ1([a1,b1])=μ2([a2,b2]), where μi are the invariant measures of fi,i=1,2. Suppose that the products of the jump ratios of Df1 and Df2do not coincide, that is, Df1 (a1 −0)/Df1 (a1 +0)⋅Df1 (b1 −0)/Df1 (b1 +0)≠Df2(a2−0)/Df2(a2+0)⋅Df2(b2−0)/Df2(b2+0) . Then the map ψ conjugating f1 and f2 is a singular function, that is, it is continuous on S1, but Dψ(x)=0 almost everywhere with respect to Lebesgue measure.