  Let $f$ and $g$ be two class $P$-homeomorphisms of the circle $S^{1}$ with
break points singularities. Assume that the derivatives $\textrm{Df}$ and
$\textrm{Dg}$ are absolutely continuous on every continuity interval of
$\textrm{Df}$ and $\textrm{Dg}$ respectively. Denote by $C(f)$ the set of break
points of $f$. For $c\in S^{1}$, denote by $\pi_{s, O_{f}(c)}(f)$ the product
of $f-$ jumps in break points lying to the $f-$ orbit of $c$ and by
$\textrm{SO}(f) = \{O_{f}(c):~c \in C(f)~\textrm{and}~\pi_{s, O_{f}(c)}(f)\neq
1\}$, called the set of singular $f$-orbits. The maps $f$ and $g$ are called
break-equivalent if there exists a topological conjugating $h$ such that
$h(\textrm{SO}(f))=\textrm{SO}(g)~~ \textrm{and} ~~ \pi_{s, O_{g}(h(c))}(g) =
\pi_{s, O_{f}(c)}(f) ~~ \textrm{for all}~~ c\in \textrm{SO}(f)$. Assume that
$f$ and $g$ have the same irrational rotation number of bounded type. We prove
that if $f$ and $g$ are not break-equivalent, then any topological conjugating
$h$ between $f$ and $g$ is a singular function i.e. it is a continuous on
$S^{1}$, but $\textrm{Dh}(x)=0$ a.e. with respect to the Lebesgue measure. As a
consequence if for some point $d\in \textrm{SO}(f)$, $\pi_{s,
O_{g}(d)}(g)\notin \{\pi_{s, O_{f}(c)}(f): c\in C(f)\}$, then the homeomorphism
conjugation $h$ is a singular function. This later result generalizes previous
results for one and two break points obtained by Dzhalilov-Akin-Temir and
Akhadkulov-Dzhalilov-Noorani. Moreover, if $f$ and $g$ do not have the same
number of singular orbits then the homeomorphism conjugating $f$ to $g$ is a
singular function.
