Continuity of the divergence of a static electric field

Question

Let $\rho:\Bbb R^3\to\Bbb R$ be a continuous charge density function. Define the electric field $\vec E:\Bbb R^3\to\Bbb R^3$ by $$\vec E(\vec r)=k\cdot\int_{\Bbb R^3}\rho(\vec{r}')\cdot\frac{\vec r-\vec r'}{\left|\vec r - \vec r'\right|^3}\,dV(\vec r'),$$ where $V$ is the volume measure and $k$ a positive constant. Does it follow that $$\operatorname{div}\vec E=D_1E^1+D_2E^2+D_3E^3$$ is continuous? If so, why?