  The article is dedicated to one of the most undeservedly overlooked
properties of the cosmological models: the behaviour at, near and due to a jump
discontinuity. It is most interesting that while the usual considerations of
the cosmological dynamics deals heavily in the singularities produced by the
discontinuities of the second kind (a.k.a. the essential discontinuities) of
one (or more) of the physical parameters, almost no research exists to date
that would turn to their natural extension/counterpart: the singularities
induced by the discontinuities of the first kind (a.k.a. the jump
discontinuities). It is this oversight that this article aims to amend. In
fact, it demonstrates that the inclusion of such singularities allows one to
produce a number of very interesting scenarios of cosmological evolution. For
example, it produces the cosmological models with a finite value of the
equation of state parameter $w=p/\rho$ even when both the energy density and
the pressure diverge, while at the same time keeping the scale factor finite.
Such a dynamics is shown to be possible only when the scale factor experiences
a finite jump at some moment of time. Furthermore, if it is the first
derivative of the scale factor that experiences a jump, then a whole new and
different type of a sudden future singularity appears. Finally, jump
discontinuities suffered by either a second or third derivatives of a scale
factor lead to cosmological models experiencing a sudden dephantomization -- or
avoiding the phantomization altogether. This implies that theoretically there
should not be any obstacles for extending the cosmological evolution beyond the
corresponding singularities; therefore, such singularities can be considered a
sort of a cosmological phase transition.
