How to prove that a function
$v_H(\mathbf{r}) = \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}d\mathbf{r'}$
is finite at any $\textbf{r}$?
$\rho(\mathbf{r'})$ appearing inside the integral is some well-behaved function that is finite everywhere, decays exponentially and vanishes at infinity.
Would a function
$v(\mathbf{r}) = \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|^n}d\mathbf{r'}$
(where $n$ is some nonnegative integer)
be also finite at any $\mathbf{r}$?