Proposition 1 ("Rolle's theorem")

Let $f$ be a real function which is finite and continuous on a closed interval $I = [a, b]$ (where $a < b$), has a derivative (finite or not) at every point of ]$a, b$[, and is such that $f(a) = f(b)$. Then there exists a point $c$ of ]$a, b$[ such that $f'(c) = 0$.

The proposition is evident if $f$ is constant: if not, $f$ takes, for example, values $> f(a)$, and so attains its least upper bound at a point $c$ interior to I (Gen. Top., IV, p. 359, th. 1). Since $f$ has a relative maximum at this point we have $f'(c) = 0$ (I, p. 20, prop. 7).