Is electric potential always differentiable?

Question

Is electric potential always differentiable?

If so, why?

If it isn't always, then what properties of a charge-distribution are required to make it differentiable?

Answer 1

One typically starts with the physically measurable electric field $\vec{E}$ and then defines the electric potential $\phi$ such that $\vec{E}$ is its derivative, so if $\phi$ weren't differentiable then it wouldn't be a very useful concept...

One exception is that it can fail to be differentiable at points where the electric field itself is not well-defined, e.g. exactly at the location of a point charge, line charge, or sheet of charge. But it will certainly be differentiable "almost everywhere".

Answer 2

The electrostatic potential might not always be differentiable. A simple theoretical example of the following is a point charge in empty space. V is differentiable everywhere except at the point where the charge is placed. Here, the function blows up and therefore becomes non-differentiable. To generalise even further, we know that $\vec E = -\nabla V$ for any conservative electric field $\vec E$. E itself may not be defined at all points in space, for various charge distributions.