Let us consider a $1D$ real function $V(x)$. When is this a classical electrostatic potential?
My take on the problem:
- $V(x)$ must be differentiable everywhere. In fact, we should be able to differentiate it $n$ times.
- $V(x)$ should vanish at $\pm \infty$.
I think these are necessary and sufficient conditions. Is this right? How do I deal with discrete charge distributions, where the potential is not differentiable at the points where the charges are present?