  Let $f_i\in C^{2+\alpha}(S^1\setminus \{a_i,b_i\}), \alpha >0, i=1,2$ be
circle homeomorphisms with two break points $a_i,b_i$, i.e. discontinuities in
the derivative $f_i$, with identical irrational rotation number $rho$ and
$\mu_1([a_1,b_1])= \mu_2([a_2,b_2])$, where $\mu_i$ are invariant measures of
$f_i$. Suppose the products of the jump ratios of $Df_1$ and $Df_2$ do not
coincide, i.e. $\frac{Df_1(a_1-0)}{Df_1(a_1+0)}\times
\frac{Df_1(b_1-0)}{Df_1(b_1+0)}\neq \frac{Df_2(a_2-0)}{Df_2(a_2+0)}\times
\frac{Df_2(b_2-0)}{Df_2(b_2+0)}$. Then the map $\psi$ conjugating $f_1$ and
$f_2$ is a singular function, i.e. it is continuous on $S^1$, but $D\psi = 0$
a.e. with respect to Lebesgue measure
