DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$

Definition

Let $I \subseteq R$ be an interval and $f : I \to R$ . Put $f^{(0)} : = f$ . Recursively, wherever $f^{(j)}$ is differentiable, put $f^{(j + 1)} : = (f^{(j)})^{'}$ , with derivatives at endpoints understood in the one-sided sense fixed by The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set and The left and right limits of $f$ at $c$ , as limits of the restrictions of $f$ to $A \cap (- \infty, c)$ and $A \cap (c, \infty)$ .

For $k \in N$ , the function is $k$ -times differentiable on $I$ if $f^{(j)}$ exists on $I$ for every $j \leq k$ . It is of class $C^{k}$ on $I$ if these derivatives exist and every $f^{(j)}$ , $0 \leq j \leq k$ , is continuous on $I$ (Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point). It is smooth, or $C^{\infty}$ , if it is $C^{k}$ for every $k \in N$ .

Since $0 \in N$ (The natural numbers $N$ (von Neumann)), $C^{0}$ means continuity. The definitions also give $C^{k + 1} \subseteq C^{k}$ . Existence of $f^{(k)}$ alone does not assert that $f^{(k)}$ is continuous.

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$

Definition

Depends on

Dependency tree · 2 levels

Sources

Higher derivatives and the classes CkCk and C∞C∞

Definition

Depends on

Dependency tree · 2 levels

Sources

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$