Consider a volume charge distribution with continuous density $\rho({\bf r'})$. The electric field at ${\bf r}$ is:

$${\bf E}({\bf r})=k\int_V \frac{\rho({\bf r'})}{R^2}\hat{\bf R}\, \mathrm dV$$

where $\bf R=|\mathbf{r}-\mathbf{r'}|$

In my textbook, the Gauss divergence theorem is applied to ${\bf E}({\bf r})$. However to apply the Gauss divergence theorem, ${\bf E}({\bf r})$ should be continuously differentiable. That is,

$$\dfrac{\partial{E_x}({\bf r})}{\partial x}, \dfrac{\partial{E_y}({\bf r})}{\partial y}, \dfrac{\partial{E_z}({\bf r})}{\partial z} \text{must be continuous.} \tag1$$

My book doesn't prove it.

So is statement $(1)$ true? If yes, how shall we prove it. If no, how can we justify the application of $GDT$?


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