Suppose $f_1, \dots, f_n$ are real-valued real analytic functions defined on an open set $B=(0,1)^d$ of $\mathbb{R}^d$. For $r \in \mathbb{R}$, set $S_r$ to be the sub-level set in $B$ defined by the system of inequalities $$f_1(x) <r, \dots, f_n(x)<r,$$ and denote by $V(r)$ the volume of the set $S_r$. It is easy to see that one generally cannot expect $V(r)$ to be a real analytic function of $r$; take, for instance, $B=(0,1)$ and $f_1(x)=x$. Nevertheless, it seem to me that if one somehow avoids "incoming" or "outgoing" conditions as $r$ varies, then $V(r)$ should depend real analytically on $r$. I would very much appreciate a reference if such a statement can be precisely formulated and proved.

Edit: The comment below by Nicolas regarding the set $B$ has been incorporated in the statement of the question.