Osgood's lemma

Osgood's lemma gives uniqueness for an initial-value problem when the vector field's modulus of continuity has a divergent reciprocal integral near zero.

Osgood uniqueness criterion: A uniqueness test requiring the reciprocal of a modulus of continuity to have a divergent integral at zero. This is the precise integral test underlying the lemma.
Ordinary differential equation: An equation relating an unknown function of one variable to its derivatives. Osgood's uniqueness conclusion applies to solutions of these equations.
Yamada–Watanabe condition: A modulus-of-continuity condition on diffusion coefficients used to establish pathwise uniqueness for stochastic differential equations. Its reciprocal-integral threshold adapts Osgood's idea to stochastic dynamics.
William Fogg Osgood: An American mathematician known for work in complex analysis and differential equations. The uniqueness criterion bears his name.
Modulus of continuity: A function measuring the maximum variation of a function between points separated by a given distance. It quantifies how rapidly the differential equation's vector field can vary.
Local Lipschitz continuity: A condition requiring function differences to be bounded locally by a constant times input differences. It is a familiar sufficient condition whose linear modulus meets Osgood's test.
Pathwise uniqueness: Uniqueness of stochastic-process solutions driven by the same noise and having the same initial state. Osgood-type conditions can secure this stronger stochastic uniqueness property.
Cauchy–Lipschitz theorem: A theorem ensuring local existence and uniqueness for differential equations with locally Lipschitz vector fields. Osgood's criterion broadens its uniqueness reach beyond Lipschitz regularity.
Bihari–LaSalle inequality: An integral inequality that bounds solutions of differential inequalities with nonlinear growth terms. It generalizes the estimate used to prevent solution differences from growing.
Hölder continuity: A regularity condition bounding function differences by a constant times a positive power of input distance. Its power-law moduli show why continuity rates matter to uniqueness.

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Hartogs's theorem on separate holomorphicityRelated: Its early formulations helped clarify which hypotheses make Hartogs's conclusion valid.