Electric field of infinite point charge at lattice on plane Infinite point charge at lattice point on $xy$ plane:

Then the electric field at $(0, 0, z)$ is calculated like this
$$\sum_{x=-\infty}^\infty \sum_{y=-\infty}^\infty \frac{z}{(x^2+y^2+z^2)^\frac{3}{2}}$$
Electric field of infinite conductor plane is constant, but is that still constant if the charge is located discretely?
 A: Good question! As Puk rightly commented, the field is not constant. But physics is usually about making the right assumptions to simplify your problem. Often one obtains what is called "intermediate asymptotics". That's exactly what the constant field of the conducting plain is: You are interested in a large-scale phenomenon, i.e. not in going to such small length scales that we can reasonably distinguish the single charged particles. At the same time we don't want to be too far away from the (finite!) conducting plate, the distance should be in a way smaller that the extent of the plate, such that the assumption of an infinite plate is reasonable. And then the modelling with a constant field is justified - and the ideal solution because we have thrown away unnecessary detail.
A: Calculations of the electric field due to a continuous, and most importantly, infinite charge distribution makes use of Gauss law, which is the best and easiest way to do it. However, if we replace the uniformly charged plate by point charges, the situation changes. Now we have point charges, we can calculate de electric field by means of the coulomb's law, but this is a bit complicated because of the vector nature of the electric field. Eventually, this can be avoided by assuming a uniform distribution of charges in the plane, but you can use the advantages of scalar nature the electrostatic potential and so you can obtain the electric field on the z-axis. In the regions close to the plane, the electric field and consequently the electrostatic potential are not uniform, however, if you explore the asymptotic behavior of those quantities, the results are similar to the obtained in the continuous case. The above can be obtained by first treating the infinite sum. I have been explored this case if you are interested (https://doi.org/10.1088/1361-6404/ac5915), but I think that, based on that, an interesting question is: how is the behavior of the electric field (or potential) assuming a non-uniform distribution of point charges? and then explore the case when the separations between point charges become close to zero.
