# Prove electric field is continuous at an external point $P$

I have a continuous charge distribution with density $$\rho$$.

Can anyone please tell how shall I prove the electric field is continuous at an external point $$P$$?

$$\displaystyle \mathbf{E}=\dfrac{1}{4 \pi \epsilon_0} \int_V \rho \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dV'$$

• Is anyone here? – N.G.Tyson May 5 '19 at 16:30
• Have you considered the differential version of Gauss' law ($\nabla \cdot \mathbf{E} = \dfrac{\rho}{\epsilon_0}$)? It means $\mathbf{E}$ is non-continuous only at places where $\rho$ is infinite. – Cuspy Code May 6 '19 at 16:59
• How? Can you provide (or provide a link) of that theorem? – N.G.Tyson May 6 '19 at 17:03
• Sorry, I don't have a rigorous proof at hand, but that's where I would start. The thing to prove would then be that a discontinuity in $\mathbf{E}$ must lead to an infinite $\nabla \mathbf{E}$. – Cuspy Code May 6 '19 at 17:15
• Then four theorems come out of it: (1) discontinuity in $\mathbf{E}$ implies infinite $\nabla \cdot \mathbf{E}$ (2) infinite $\nabla \cdot \mathbf{E}$ implies discontinuity (3) continuity in $\mathbf{E}$ implies finite $\nabla \cdot \mathbf{E}$ (4) finite $\nabla \cdot \mathbf{E}$ implies continuity in $\mathbf{E}$. Can you give at least a mathematical or "physics" proof of the last one? – N.G.Tyson May 6 '19 at 17:21