DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-02

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.

Equicontinuity, pointwise boundedness, and uniform boundedness for families in $C (K, R)$

Definition

Let $(K, d)$ be a nonempty compact metric space and let $F \subseteq C (K, R)$ in the sense of The space $C (K, R)$ of continuous real-valued functions on a nonempty compact metric space. The family $F$ is equicontinuous at $a \in K$ when, for every $ε > 0$ , there is $δ > 0$ such that for every $f \in F$ and every $x \in K$ , $d (x, a) < δ$ implies $∣ f (x) - f (a) ∣ < ε$ . It is equicontinuous when this holds at every $a \in K$ .

It is pointwise bounded when, for every $a \in K$ , the set ${f (a) : f \in F}$ is bounded in $R$ . It is uniformly bounded when a real $M \geq 0$ satisfies $∣ f (x) ∣ \leq M$ for every $f \in F$ and $x \in K$ .

Depends on

The space $C (K, R)$ of continuous real-valued functions on a nonempty compact metric space

Dependency tree · 2 levels

Sources

The Ascoli--Arzelà Theorem (MIT)

Equicontinuity, pointwise boundedness, and uniform boundedness for families in C(K,R)C(K,R)

Definition

Depends on

Dependency tree · 2 levels

Sources

Equicontinuity, pointwise boundedness, and uniform boundedness for families in $C (K, R)$