Suppose that we have $u \in C^1(\Omega)$, for an open set $\Omega \subset \mathbb{R}^n$. I want to show that $|u|$ is differentiable almost everywhere. I intuitively understand that this is true, but I need to prove it.
I need it to use the fundamental theorem of calculus of Lebesgue for the partial derivatives of $|u|$.
$A=${$\Omega$: $u >0$}
$B=${$\Omega$: $u \le 0$}
since $f$ is continuous, $A$ is actually a countable union of disjoint intervals and a finite number of points,$B$ is also a countable union of intervals and a finite number of points. $A$ and $B$ are disjoint.
So You can use absolute continuity and differentiabilty on the intervals of $A$ and $B$ separately. The finite number of points have measure zero.