Before getting into the question, here are some remarks:
Given a single point charge, the value of electric field at the position of the charge is singular/undefined, which makes sense, since a particle cannot interact with itself.
Given a discrete charge distribution, the value of the field at an empty point(i.e. no particles reside at that point), is the field contribution from all the charges. However, if the point of interest contains a point-charge, then the value of the field is the field contribution from all the charges except the charge that resides at that point.
Now here's my inquiry: In the case of discrete charge distribution, it makes sense to speak of the value of the electric field at a given point, whether that point contains a point-charge or not. What about continuous distributions?
My intuition says yes, we can; since one can think of continuous distributions as an extension of discrete ones, with the difference being that the former contains uncountably infinite charges. Therefore, to calculate the field value at any given point on the continuous charge distribution, one has to consider the field contributions from all the charges except the one residing at that point.
Example: The value of the electric field at any point on an infinite sheet (plane) of charges should be zero. Since at any given point on the sheet, one can think of that point as being surrounded by infinite concentric rings, where the field contribution from each ring (by symmetry) is zero.
So does it make sense to speak of the value of electric field on continuous charge distribution? Or is it not defined?