Corollary

Let $f$ be a real function which is finite and continuous on $[a, b]$ (where $a < b$), and has a derivative (finite or not) at every point. Then there exists a point $c$ of ]$a, b$[ such that $f(b) - f(a) = f'(c)(b - a)$.

We need only apply prop. 1 to the function $f(x) - \frac{f(b) - f(a)}{b - a}(x - a)$.

This corollary signifies that there is a point $M_c = (c, f(c))$ on the graph $C$ of $f$ such that $a < c < b$ and such that the tangent to $C$ at this point is parallel to the line joining the points $M_a = (a, f(a))$ and $M_b = (b, f(b))$.