[noobermin]

I mean, the standard model says they are fundamental, although there are theories that they may be "made up" of other things like in string theory, although those theories have (imo) struggled to compare to experiment or demand experiments that are infeasible today. Regarding "continuous and discrete", quantum mechanically, electrons aren't definitein space, you might be referring to that. In plasma physics we sit above the quantum limit (neglecting p and x variance or more correctly, the variances' product well exceed \hbar), so we treat them discretely, so in a sense, as all classical physics is, it's an approximation too.

My point is that even in classical physics, I usually don't care about fluctuations on small scales which will be noisy, so on top of the classical approximation I make another approximation where I replace a noisy function with a smoother function that well approximates the noisy one. Smoothing out the noise is an important tool for theoretical understanding (as OP's student pointed out here), but it's important to remember it's just an approximation.

EDIT: re the other replier. Another example is I treat ions as "fundamental" too, as we don't reach energies and conditions where their constituent nuclei matter, only ionization.