Corollary: Estimating the Growth of a Function with its Derivative

(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

$m(y-x)\le f(y)-f(x)\le M(y-x).$

Proofs: 1 Corollaries: 1

Sections: 1

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983