I've a simple question concerning the perimeter of level sets of a smooth function.
Let $f:\Omega \to \mathbb{R}$ be a smooth function defined on a bounded domain of $\mathbb{R}^n$. We set $A_s:=\{f>s\}$ for any $s \in \mathbb{R}$. By the Morse-Sard lemma, we know that for a.e. $s \in \mathbb{R}$, $\partial A_s$ is a smooth hypersurface. We denote by $I$ the set of such values $s$ and by $\mathcal{P}(A_s)$ the perimeter of $A_s$ for any $s \in I$. Does there exist C>0 such that: $$ |\mathcal{P}(A_s) - \mathcal{P}(A_{s'})|\le C|s - s'| \qquad \forall s \in I? $$
I've looked into Federer's book on Geometric Measure Theory, without any success, and I'm looking for other references in which I could find whether this question has been answered yet.
Thanks.