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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 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.
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.
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.
When X is a vector space and the two metrics and are those induced by norms ‖ ‖ and ‖ ‖, respectively, then strong equivalence is equivalent to the condition that, for all , ‖ ‖ ‖ ‖ ‖ ‖ For linear operators between normed vector spaces, Lipschitz continuity is equivalent to continuity—an operator satisfying either of these ...
For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a hypersurface in this case), but can be every intersection of the graph itself with a hyperplane (in the case of functions of two ...
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.