  A regularization procedure, that allows one to relate singularities of
curvature to those of the Einstein tensor without some of the shortcomings of
previous approaches, is proposed. This regularization is obtained by requiring
that (i) the density $|det {\bf g}|^{{1/2}}G^a_{b}$, associated to the Einstein
tensor $G^a_{b}$ of the regularized metric, rather than the Einstein tensor
itself, be a distribution and (ii) the regularized metric be a continuous
metric with a discontinuous extrinsic curvature across a non-null hypersurface
of codimension one. In this paper, the curvature and Einstein tensors of the
geometries associated to point sources in the 2+1-dimensional gravity and the
Schwarzschild spacetime are considered. In both examples the regularized
metrics are continuous regular metrics, as defined by Geroch and Traschen, with
well defined distributional curvature tensors at all the intermediate steps of
the calculation. The limit in which the support of these curvature tensors
tends to the singular region of the original spacetime is studied and the
results are contrasted with the ones obtained in previous works.
