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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 ...
Met is not the only category whose objects are metric spaces; others include the category of uniformly continuous functions, the category of Lipschitz functions and the category of quasi-Lipschitz mappings. The metric maps are both uniformly continuous and Lipschitz, with Lipschitz constant at most one.
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.
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 ,
In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function. The term is named after the German mathematician Rudolf Lipschitz.
This also includes β = 1 and therefore all Lipschitz continuous functions on a bounded set are also C 0,α Hölder continuous. The function f(x) = x β (with β ≤ 1) defined on [0, 1] serves as a prototypical example of a function that is C 0,α Hölder continuous for 0 < α ≤ β, but not for α > β.
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