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

How statement and proof provenance work

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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 $C(K,\mathbb{R})$ of continuous real-valued functions on a nonempty compact metric space

Used by

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Ascoli--Arzelà Theorem (MIT) (standard reference, not scraped)

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