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I can only offer a partial answer to the first portion of your question. one comes across functions that diverge at a given point, typical examples would be the Coulomb or the gravitational forces being ∝ 1/r², clearly they diverge at r=0... Gravitational force isn't actually proportional to 1/r². Take a look at the plot of gravitational potential on ...


I can't speak to singularities in the sense of general relativity, but your example of $1/r^2$ laws in classical physics is actually solved most of the time by internal structure. One of my physics professors used to always say that nature solves infinities with internal structure. For example, for a charged sphere of uniform charge density, the electric ...

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