I need to read this more carefully, but a discontinuity in the derivatives of a system of equations at an interface is fairly common. If there are two regions obeying Maxwell's equations with a difference in the derivatives of the E/B fields at the boundary, this means the boundary has some delta function of charge or currents on the interface.

Likewise in GR. There is a whole sub-field of joining solutions to Einstein equations together and determining the material at the interface. Wormhole solutions can be made with this cut-and-paste approach. (The best book on this is Eric Poisson's "A Relativist's Toolkit").

The paper referenced here is 200+ pages. The abstract states in part "We prove that for all such data, the maximal Cauchy evolution can be extended across a non-trivial piece of Cauchy horizon as a Lorentzian manifold with continuous metric.", which is what disproves strong cosmic censorship.

I have no grasp of the details (my PhD in GR junction conditions is from the 90's and I left the field).