Let $f:\mathbb{R}\rightarrow \mathbb{R}$. If $f$ is differentiable I can write $$ \int_a^bf'(x)dx \leq |b-a|\sup_{x\in[a,b]} |f'(x)|. $$
I believe that a similar inequality should hold when $f$ is only absolutely continuous so that it is only differentiable almost everywhere. But in that case I'm not sure how to think of $\sup_{x\in[a,b]} |f'(x)|$. The function might not be differentiable at some points in $[a,b]$ but that should not affect the integral. Could I say that the $\sup$ is over the points where $f$ is differentiable? Is there a better way to write this?