Lipschitz continuity

A function is Lipschitz continuous if one finite constant bounds every output difference by that constant times the corresponding input distance. This uniform bound limits how sharply the function can vary.

Lipschitz constant: A nonnegative constant that bounds a function’s output differences by input distances. It is the smallest uniform bound when the least such constant exists.
Continuity: A function is continuous at a point when sufficiently small input changes produce arbitrarily small output changes there. Every Lipschitz function is continuous, but continuity alone gives no linear rate.
Banach fixed-point theorem: A contraction on a complete metric space has a unique fixed point, and iteration converges to it. Its contraction hypothesis is a strict Lipschitz bound.
Square-root function: The function mapping each nonnegative real number to its nonnegative square root. It is continuous at zero but fails every finite Lipschitz bound on intervals containing zero.
Metric space: A set equipped with a distance function satisfying nonnegativity, symmetry, identity, and triangle inequality. The Lipschitz inequality is defined using distances in the domain and codomain.
Absolute value: The absolute value of a real number is its nonnegative distance from zero. For real functions, the defining inequality uses absolute differences in both input and output.
Picard–Lindelöf theorem: A theorem guaranteeing local existence and uniqueness for ordinary differential equations with a locally Lipschitz state dependence. Local Lipschitz bounds prevent distinct solution trajectories from branching.
Weierstrass function: A classical continuous function constructed to be nowhere differentiable. Its extreme oscillation contrasts with the regularity imposed by a Lipschitz bound.
Uniform continuity: A function is uniformly continuous when each output tolerance has one input tolerance that works throughout its domain. A Lipschitz bound directly supplies the uniform input tolerance.
Norm (mathematics): A norm assigns a nonnegative length to vectors and satisfies homogeneity and the triangle inequality. Vector-valued Lipschitz bounds commonly measure differences with norms.

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