Theorem: Darboux's Theorem

Let $a < b,$ $[a,b]$ be a closed real interval, and $f:[a,b]\to\mathbb R$ a continuous function. Further, let $f$ be differentiable on the open real interval $]a,b[.$ Then there is an intermediate value $\xi\in ]a,b[$ with $$f'(\xi)=\frac{f(b)-f(a)}{b-a}.$$

From the geometrical point of view, this intermediate value theorem states that the gradient of the secant through the points $(a,f(a))$ and $(b,f(b))$ equals the gradient of the tangent of the graph of $f$ on some intermediate point $(\xi,f(\xi))$.

This theorem is named after Gaston Darboux (1842 - 1917).

Proofs: 1 Corollaries: 1

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