Scope, endpoint, factorial, and deferred-remainder conventions

Higher derivatives are the recursive objects of Higher derivatives and the classes $C^k$ and $C^\infty$. Darboux's theorem (Darboux's theorem: every derivative has the intermediate-value property) concerns every first derivative, without assuming that derivative continuous. The two L'Hôpital theorems, L'Hôpital's rule for the $0/0$ form at finite or infinite, one-sided endpoints and L'Hôpital's rule for the $\infty/\infty$ form at finite or infinite, one-sided endpoints, require their stated derivative and nonvanishing hypotheses and do not assert converses.

Endpoint derivatives and finite-endpoint limits are one-sided when the domain supplies only one side. Natural factorials and binomial coefficients (The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$, The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$) enter real formulas through the canonical embedding $\iota$ of The canonical natural $\iota(n) = n \cdot 1_F$ of a field. Darboux's property alone has no general continuity converse; the page's An injective or monotone derivative on an interval is continuous proves continuity under the stated injectivity or monotonicity hypotheses.

The Schlömilch-Roche formula (Taylor's Schlömilch–Roche remainder formula) assumes an $(n+1)$-st derivative on an interval. Peano's formula (Peano's form: the normalized Taylor remainder tends to zero) assumes $n$-fold differentiability on an open interval $N_\delta(a)$ around the expansion point (The $\varepsilon$-neighbourhood and the punctured $\varepsilon$-neighbourhood of a point of $\mathbb{R}$), but not continuity of the $n$-th derivative. No integral remainder, Borel interpolation theorem, or assertion about Dini derivatives is made here.