  We consider special flows over the rotation on the circle by an irrational
$\alpha$ under roof functions of bounded variation. The roof functions, in the
Lebesgue decomposition, are assumed to have a continuous singular part coming
from a quasi-similar Cantor set (including the Devil's staircase case).
Moreover, a finite number of discontinuities is allowed. Assuming that $\alpha$
has bounded partial quotients, we prove that all such flows are weakly mixing
and enjoy weak Ratner's property. Moreover, we provide a sufficient condition
for the roof function to obtain a stability of the cocycle Ratner's property
for the resulting special flow.
