  The rigidity theory for circle homeomophisms with breaks was studied
intensively in the last 20 years.
  It was proved that under mild conditions of the Diophantine type on the
rotation number any two $C^{2+\alpha}$ smooth circle homeomorphisms with a
break point are $C^1$ smoothly conjugate to each other, provided that they have
the same rotation number and the same size of the break.
  In this paper we prove that the conjugacy may not be $C^{2-\nu}$ even if the
maps are analytic outside of the break points.
  This result shows that the rigidity theory for maps with singularities is
very different from the linearizable case of circle diffeomorphisms where
conjugacy is arbitrarily smooth, or even analytic, for sufficiently smooth
diffeomorphisms.
