I'm a mathematician self-studying physics for fun and I'm trying to wrap my head around a simple concept, which is a consequence of approximations. Basically, by Coulomb's law, if I have a point charge and look at the electric field, then the electric field grows without a bound when I come close enough to the charge. On the other hand, if I have a flat infinite plane with a continuous uniform surface charge, then the electric field is uniform and independent of the distance to the plane and is discontinuous at the boundary. Ultimately, all charge is discrete and if we distribute discrete point charges on a plane, then obviously the magnitude of the electric field again grows without bound again when we come close enough to one the point charges.
Now my issue with some of the arguments in e.g. Griffith's electrodynamics textbook is that some results can be somewhat easily proven by just saying that "yeah, everything is actually discrete, so just apply superposition", which sometimes avoids some pesky integrals. On the other hand, as the example above shows, we can get contradicting results between the continuous and discrete cases.
My somewhat fuzzy question is mainly the following: When should one be careful of suddenly making conclusions using discrete arguments when otherwise working in the macro scale assuming continuous charge distributions?
I assume part of not losing sleep over these things is learning to think like a physicist.