Mean value estimate

Mean value estimate lemma: Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If there exists $M\ge 0$ such that $|f'(x)|\le M$ for all $x\in(a,b)$, then

This is an immediate quantitative consequence of the mean value theorem and is a standard step in results like bounded derivative implies uniform continuity .