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 alternatively they can allow for the extension of geodesics through them. In the latter case only the components of the Riemann tensor representing tidal forces diverge. A subclass of these soft singularities, the Sudden Future Singularity (SFS) occurs at finite time, finite scale factor and finite Hubble parameter, only the deceleration parameter being divergent. In a Friedmann universe evolving in the framework of general relativity they are realized by perfect fluids with regular energy density and diverging pressure 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 any matter (in particular the simplest, the dust) to these models, an unwanted feature appears: at the finite scale factor of the SFS the matter energy density remains finite, implying (for a spatially flat universe) a finite Hubble parameter, hence finite expansion rate, rather then full stop. The universe would then 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 which are continuous across the SFS are then 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 this is no disadvantage as they anyhow diverge at the singularity.

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