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In fact, spacetime behaves very well at singularities, in the old framework of General Relativity. Singularities occur, indeed, as Penrose and Hawking proved long time ago. But physics and differential geometry can be done there. Equivalent versions of Einstein's equation can be written. They give the same result as the standard Einstein equation where there are not singularities, but apply also where there are singularities. The singularities don't destroy information.
The stationary black holes admit coordinate systems which makes the singularity of the metric to be "benign", i.e. smooth and without infinities:
Schwarzschild Singularity is Semi-Regularizable
Analytic Reissner-Nordstrom Singularity
Kerr-Newman Solutions with Analytic Singularity and no Closed Timelike Curves
The FLRW Big Bang singularity is already benign:
Big Bang singularity in the Friedmann-Lemaitre-Robertson-Walker spacetime
Beyond the Friedmann-Lemaitre-Robertson-Walker Big Bang singularity
For benign singularities one can do differential geometry
On Singular Semi-Riemannian Manifolds
Warped Products of Singular Semi-Riemannian Manifolds
Cartan's Structural Equations for Degenerate Metric
and write equivalent to Einstein's equation
Einstein equation at singularities
A large class of benign singularities also answer one big puzzle of Penrose:
On the Weyl Curvature Hypothesis
The singularities are compatible with global hyperbolicity and don't destroy information, if we know how to continue the equations beyond the singularities: