Consider a function $f:(x_0,x_1,...,x_n)\in\mathbb{R}^n\rightarrow f(x_0,...,x_n)\in\mathbb{R}$. $f$ is a ratio of polynomials in $x_0,...,x_n$ which only has simple poles in the variable $x_0$, whose position might depend on the value of the other $n-1$ variables, $x_1,...,x_n$ (however, these poles never overlap for any value of such variables). Away from the simple poles, it is $C^{\infty}$. I'm trying to understand the principal value
\begin{equation} g(x_1,...,x_n)=\operatorname{PV}\int dx_0 f(x_0,...,x_n), \end{equation} and I'm wondering if any general property of $g$ can be deduced, and specifically I would like to know: can it be stated that $g$ has no step discontinuities in any of the variables $x_1,...,x_n$? If there are not enough assumptions, what could be added to make the statement true?