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>>qqqqqu+(OP)
When two particles get closer, their mutual gravitational attraction increases. As the distance approaches zero, the force approaches infinity. In the limit of d -> 0, the energy released -> infinity. Obviously at some scale the notion of a point mass breaks down, but even quantum theory would be problematic if we think of a wave function as describing a probability distribution, wouldn't it? What's the "official" story on this?

>>bhk+An
What follows is very hand-wavy, and the renormalization sibling post may touch on it.

An answer is that the d->0 approaches infinity presumes a nice, continuous analytic function. If d->epsilon, you can't get to that singularity.

There was an equivalent problem in the E/M space with "The Ultraviolet Catastrophe" [1], which turned out to go away if you assumed quantization.

I'm not going to claim this is a perfect analog to the gravity problem, only that a lot of physics doesn't quite work right when you assume continuity. (The Dirac delta is a humorous exception that proves the rule here, in that doing the mathematically weird thing actually is closer to how physics works, and it required "distribution theory" as a discipline to prove it sound.)

[1] https://en.wikipedia.org/wiki/Ultraviolet_catastrophe