# Doubt in application of $GDT$ in electrostatics

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 $$GDT$$?