Mean value theorem

For real functions continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $\xi \in (a,b)$ withApply Rolle's theorem to a linear combination of the functions with equal endpoint values. If $g^{\prime }$ never vanishes, $g(b)\ne g(a)$ and the equality can be divided into a ratio. This form directly proves the zero-over-zero endpoint case of L'Hôpital's rule, without continuity of the derivatives.

If $f$ is continuous at an interval endpoint, differentiable in its interior, and has a finite one-sided derivative there, then its secant slope and tangent slope are arbitrarily close at some interior point. Apply the mean value theorem on a short endpoint interval; both the secant at its far endpoint and the secant at the resulting intermediate point approach the same one-sided derivative. No continuity of the derivative is required.