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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.

If a real function on an interval obeys $|f(x)-f(y)|\le C|x-y{|}^{\alpha }$ for finite $C$ and $\alpha >1$, its difference quotients tend to zero everywhere. The mean value theorem makes it constant. Equivalently, divide a segment into $N$ pieces to obtain $|f(x)-f(y)|\le C|x-y{|}^{\alpha }{N}^{1-\alpha }\to 0$.

If $f$ is Riemann integrable and $g$ is Lipschitz on an interval containing the range of $f$, then $g\circ f$ is Riemann integrable. With Lipschitz constant $K$, the oscillation on each partition cell satisfies $\mathrm{osc}(g\circ f)\le K\mathrm{osc}(f)$, and hence the gap between Darboux sums is multiplied by at most $K$. A continuously differentiable $g$ is Lipschitz on each closed bounded interval by the mean value theorem and the extreme value theorem applied to $g^{\prime }$.