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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.
In 1963, Ralph DeMarr showed that if and are both Lipschitz continuous, and if the Lipschitz constant of both is , then and will have a common fixed point. [6] Gerald Jungck refined DeMarr's conditions, showing that they need not be Lipschitz continuous, but instead satisfy similar but less restrictive criteria.
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 ,
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.
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.
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.
Eilenberg's inequality, also known as the coarea inequality is a mathematical inequality for Lipschitz-continuous functions between metric spaces. Informally, it gives an upper bound on the average size of the fibers of a Lipschitz map in terms of the Lipschitz constant of the function and the measure of the domain.