  We consider the metrics of the General Relativity, whose energy-momentum
tensor has a bounded support where it is continuous except for a finite step
across the corresponding boundary surface. As a consequence, the first
derivative of the metric across this boundary could perhaps present a finite
step too. However, we can assume that the metric is ${\cal C}^1$ class
everywhere. In such a case, although the partial second derivatives of the
metric exhibit finite (no Dirac $\delta$ functions) discontinuities, the Dirac
$\delta$ functions will still appear in the conservation equation of the
energy-momentum tensor. As a consequence, strictly speaking, the corresponding
metric solutions of the Einstein field equations can only exist in the sense of
distributions. Then, we assume that the metric considered is ${\cal C}^1$ class
everywhere and is a solution of the Einstein field equations in this sense. We
explore the consequences of these two assumptions, and in doing so we derive
the general conditions that constrain the jumps in the second partial
derivatives across the boundary. The example of the Oppenheimer-Snyder metric
is considered and some new results are obtained on it. Then, the formalism
developed in this exploration is applied to a different situation, i.e., to a
given generalization of the Einstein field equations for the case where the
partial second derivatives of the metric exist but are not symmetric.
