Let $f: I \to \mathbb R$ be a differentiable function, where $I$ is an open interval, and assume that there exists a constant $C>0$ such that for all $x \in I$, $|f'(x)| \leq C$.
Note that I am not assuming that $f$ is of class $C^1$: so $f'$ is a priori not continuous or even Riemann-integrable, but $|f'|$ is bounded and positive and Lebesgue-integrable.
Question: is it true that for all $x,y \in I$, we have $|f(y)-f(x)| \leq \int_x^y |f'(t)| dt$?
Using the Mean Value Theorem, you could express $f(x)-f(y)$ as a Riemann sum of $f'$, but since a priori $f'$ is not Riemann integrable I'm not sure this helps. I guess the question probably amounts to knowing the sharp regularity conditions on $f$ to have $f(y)-f(x)=\int_x^y f'(t) dt$.
