Darboux's Theorem

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 Jean Gaston Darboux.