integral over almost sure existing derivatives

integral over almost sure existing derivatives

Let $f$ and $g$ be two real-valued Lipschitz functions, with Lipschitz
constant smaller or equal to 1, on $[a,b]$ whose derivatives exist only
$\lambda$-almost surely. Does then the following hold: \begin{align*}
\int_r^t|\frac{d}{dx}f(x)-\frac{d}{dx}g(x)|dx \leq t-r \end{align*} I was
thinking of \begin{align*} \int_r^t|\frac{d}{dx}f(x)-\frac{d}{dx}g(x)|dx
=\int_A\frac{d}{dx}f(x)-\frac{d}{dx}g(x)dx
+\int_B\frac{d}{dx}g(x)-\frac{d}{dx}f(x)dx \end{align*} with
$A=\{x:\frac{d}{dx}f(x)-\frac{d}{dx}g(x)\geq 0\}$ and $B=A^c$, but I am
confused about the almost sure existence of the derivatives.