  Both dark energy models and modified gravity theories could lead to
cosmological evolutions different from either the recollapse into a Big Crunch
or exponential de Sitter expansion. The newly arising singularities may
represent true endpoints of the evolution or allow for the extension of
geodesics through them. In the latter case the components of the Riemann tensor
representing tidal forces diverge. Sudden Future Singularities (SFS) occur at
finite time, finite scale factor and finite Hubble parameter, only the
deceleration parameter diverges. The energy density of a perfect fluid is
regular and its pressure diverges at the SFS. A particular SFS, the Big Brake
occurs when the energy density vanishes and the expansion arrives at a full
stop at the singularity. Such scenarios are generated by either a particular
scalar field (the tachyon field) or the anti-Chaplygin gas. By adding matter to
these models, an unwanted feature appears: at the finite scale factor of the
SFS the energy density of matter remains finite, implying (for a spatially flat
universe) a finite Hubble parameter, hence finite expansion rate, rather then
full stop. The universe would further expand through the singularity, this
nevertheless seems forbidden as the energy density of the tachyonic field /
anti-Chaplygin gas would become ill-defined. This paradox is relieved in the
case of the anti-Chaplygin gas by redefining its energy density and pressure in
terms of distributions peaked on the singularity. The regular cosmological
quantities, continuous across the SFS are the energy density and the square of
the Hubble parameter; those allowing for a jump at the SFS are the Hubble
parameter and expansion rate (both being mirror-symmetric). The pressure and
the decelaration parameter will contain Dirac delta-function contributions
peaked on the SFS, however they anyhow would diverge at the singularity.
