Motivated by the question Compute derivative of $g(x)=\mathcal L(\{y \in [a,b]: x >f(y)\})$, I'd like to ask:
Let $f:\mathbb{R} \to \mathbb{R}$. How can one compute the limit $$\lim_{h \to 0^+} \frac{\mathcal L(f^{-1}([x,x+h))}{h},$$ where $f^{-1}$ denotes the pre-image of $f$; $\mathcal L$ the Lebesgue measure; and assuming only that $f$ is Lipschitz continuous?
In the special case $f$ monotone, the question is much easier, since $$f^{-1}([x,x+h)) = [f^{-1}(x), f^{-1}(x+h)).$$