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For a Lipschitz continuous function, there exists a double cone (white) whose origin can be moved along the graph so that the whole graph always stays outside the double cone. In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions.
The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for k ≤ 1, then the mapping is said to be a non-expansive map. More generally, the idea of a contractive mapping can be defined for maps between metric spaces.
Such functions are always continuous functions. They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive maps , nonexpanding maps , weak contractions , or short maps . Specifically, suppose that X {\displaystyle X} and Y {\displaystyle Y} are metric spaces and f {\displaystyle f} is a function from X {\displaystyle X} to ...
In category theory, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms. This is a category because the composition of two metric maps is again a metric map. It was first considered by Isbell (1964).
The distinction between independent contractor and employee is an important one in the United States, as the costs for business owners to maintain employees are significantly higher than the costs associated with hiring independent contractors, due to federal and state requirements for employers to pay FICA (Social Security and Medicare taxes) and unemployment taxes on received income for ...
The limit function is also Lipschitz continuous with the same value K for the Lipschitz constant. A slight refinement is A slight refinement is A set F of functions f on [ a , b ] that is uniformly bounded and satisfies a Hölder condition of order α , 0 < α ≤ 1 , with a fixed constant M ,
The particular case = is referred to as Lipschitz continuity. That is, a function is Lipschitz continuous if there is a constant K such that the inequality ((), ()) (,) holds for any ,. [15] The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem concerning the solutions of ordinary differential equations.
Rademacher's theorem is a special case, due to the fact that any Lipschitz function on Ω is an element of the space W 1,∞ (Ω). [9] There is a version of Rademacher's theorem that holds for Lipschitz functions from a Euclidean space into an arbitrary metric space in terms of metric differentials instead of the usual derivative.