(related to Theorem: Darboux's Theorem)
Let $a < b,$ $[a,b]$ be a closed real interval, and $f:[a,b]\to\mathbb R$ a continuous function, which is differentiable on the open real interval $]a,b[.$ Furthermore, let $f'$ be bounded from below and from above by some constants $m,M\in\mathbb R$ with
$m\le f'(\xi)\le M$ for all $\xi\in]a,b[.$
Then for all $x,y$ with $a\le x\le y\le b$, the difference $f(y)-f(x)$ can be estimated by
$m(y-x)\le f(y)-f(x)\le M(y-x).$
Sections: 1