Lipschitz continuity

The concept of Lipschitz continuity is well-defined on metric spaces.

Clearly, this also means that boundedness is no longer equivalent to Lipschitz continuity in this context.

While Lipschitz gave his name to the Lipschitz continuity condition, he worked in a broad range of areas.

A special type of Lipschitz continuity, called contraction, is used in the Banach fixed point theorem.

A generalization of Lipschitz continuity is called Hölder continuity.

The signum function does not satisfy the Lipschitz continuity condition required for the usual theorems guaranteeing existence and uniqueness of strong solutions.

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.

Lipschitz continuity, a smoothness condition for functions that is stronger than ordinary continuity (named after mathematician Rudolph Lipschitz).

In the theory of differential equations, Lipschitz continuity is the central condition of the Picard-Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem.

It requires Lipschitz continuity, while the Peano theorem requires only continuity; but it proves both existence and uniqueness where the Peano theorem proves only the existence of solutions.

The Hirzebruch-Riemann-Roch theorem, Lipschitz continuity, the Petri net, the Schönhage-Strassen algorithm, Faltings' theorem and the Toeplitz matrix are all named after University of Bonn mathematicians.

Riemannian manifolds are special cases of metric spaces, and thus one has a notion of Lipschitz continuity, Hölder condition, together with a coarse structure, which leads to notions such as coarse maps and connections with geometric group theory.