Electrodynamics makes heavy use of vector calculus, which in turn is about differentiation and integration of scalar and vector fields in $\mathbb{R}^3$. At this point everything seems fine to me, since the physical space is isomorphic to $\mathbb{R}^3$. However, Maxwell's laws rely on mathematical abstractions such as supposing that electric charge is a continuous variable.
For instance: $$\nabla \cdot \mathbf{D}=\rho$$ In the LHS, there is a local quantity, since the divergence of a vector field is defined at every point of space. On the other hand, in the RHS there is a global quantity, i.e., it can only be defined as a mean charge density in volumes much greater than the dimensions of the charge carriers involved. This is because electric charge is quantized. All charged particles have charges that are integer multiples of $\frac{1}{3}e$. Since the charge carriers are generally very small, macroscopically it seems that we can indeed define $\rho$ at every point of space, even though this statement does not make much mathematical sense. For instance, in a point of "empty" space (a point not belonging to the charge carriers), $\rho=0$.
Even though I am aware of the extraordinarily high predictive power of classical electrodynamics, the assumption of considering charges continuous instead of discrete seems pretty far-fetched. I'd like to know if, besides quantum applications, Maxwell's laws show some kind of error due to this assumption.