DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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Moduli of continuity and the Osgood divergence condition

Definition

A modulus of continuity is a continuous nondecreasing function $ρ : [0, r_{0}] \to [0, \infty)$ with $ρ (0) = 0$ and $ρ (r) > 0$ for $r > 0$ . A vector field has state modulus $ρ$ on a set if

$∥ F (t, x) - F (t, y) ∥_{2} \leq ρ (∥ x - y ∥_{2})$

whenever the two points with the same time lie in that set and $∥ x - y ∥_{2} \leq r_{0}$ . Equivalently, the phrase is used without that last qualifier only on sets whose state-space diameter is at most $r_{0}$ .

The modulus satisfies the Osgood divergence condition when

$\int_{0^{+}}^{r_{0}} \frac{d r}{ρ (r)} = + \infty,$

meaning that the compact truncation integrals $\int_{ε}^{r_{0}} d r / ρ (r)$ are unbounded as $ε ↓ 0$ . The value at $0$ is never divided by.

Depends on

Used by

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Sources

Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, Ch. 2 (standard reference, not scraped)
Jiri Lebl, Basic Analysis I, Section 6.3 (standard reference, not scraped)