point charge problem

The concept of a point charge is well known to have some problems.
Suppose we consider a point charge to be really a spherically
symmetric charge distribution of radius r equal to epsilon with
epsilon having the property that epsilon squared is greater than zero
and and any higher power of epsilon is equal to zero.
This would seem to describe a sphere that is not the boundary of a
ball since it encloses no volume.
Aside from the fact that this sounds weird, can anyone explain to me
what kind of difficulties this approach to the point charge problem
has that other approaches don't? Conversely, it does seem to address
at least the problem that a volume (or surface) integral containing
this object would yield a finite charge even though right at the
location of the charge the density would be infinite (since the volume
at the location of the charge is zero), or am I mistaken?

Thanks,

Armin

.

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