On mild mixing of special flows over irrational rotations under piecewise smooth functions

It is proved that all special flows over the rotation by an irrational $\alpha$ with bounded partial quotients and under $f$ which is piecewise absolutely continuous with a non-zero sum of jumps are mildly mixing. Such flows are also shown to enjoy a condition which emulates the Ratner condition introduced in \cite{Rat}. As a consequence we construct a smooth vector--field on $\T^2$ with one singularity point such that the corresponding flow $(\phi_t)_{t\in\R}$ preserves a smooth measure, its set of ergodic components consists of a family of periodic orbits and one component of positive measure on which $(\phi_t)_{t\in\R}$ is mildly mixing and is spectrally disjoint from all mixing flows.