What are some conditions that ensure that a function $f(x) : \mathbb{R} \to \mathbb{R}$ which is in $L^1_{loc}$ and almost everywhere differentiable (in the classical sens ) with derivative in $L^1_{loc}$ has its derivative equal to its weak derivative (its derivative in the sens of distributions) ie :
$$ \forall \phi \in C^{\infty}_c(\mathbb{R}) ; \int f'\phi = – \int f \phi'$$
For example this is false for the characteristic function $\xi_{[0;1]} $which weak derivative is the difference of two dirac measures while its classical derivative is almost everywhere 0.
It works on the other hand for $C^1$ functions.
Does it work for Lipschitz functions ? Are there some necessary conditions ?
Edit : It does work for lipshitz function thx to the dominated convergence theorem
Best Answer
The necessary and sufficient condition is that $f$ should be absolutely continuous. This is a version of the fundamental theorem of calculus, and can be found in most graduate-level real analysis books.