Existence and continuity of pressure in classic statistical physics

Let $U(y)$ be a potential such that
\begin{gather*} U(y)\ge\psi(y),\quad0\le y\le a,\quad\psi(y),\ y^r\to\infty,\quad y\to0, \\ |U(y)|\le C|y|^{-(r+\varepsilon)},\quad y>a,\quad\varepsilon>0,\quad C<\infty \end{gather*}
where $\psi(y)$ is a monotone function. Let $\frac{|\Omega_N|}N\to v$, $0<v<\infty$, where $|\Omega_N|$ is the volume of an $r$-dimensional cube $\Omega_N$, and put
$$ f(v,\beta)=\lim_{N\to\infty}\frac1N\ln\int_{\Omega_N}\dots\int_{\Omega_N}\exp\biggl\{-\beta\sum_{i\ne j}U(|x_i-x_j|)\biggr\}dx_1\dots dx_N. $$
It is proved that $\frac{\partial f(v,\beta)}{\partial v}$ exists and is continuous.