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Concept
Lipschitz continuity
Summary
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity). For instance, every function that is defined on an interval and has bounded first derivative is Lipschitz continuous.
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. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem.
We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line:
Continuously differentiable ⊂ Lipschitz continuous ⊂ -Hölder continuous,
where . We also have
Lipschitz continuous ⊂ absolutely continuous ⊂ uniformly continuous.
Given two metric spaces (X, dX) and (Y, dY), where dX denotes the metric on the set X and dY is the metric on set Y, a function f : X → Y is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all x1 and x2 in X,
Any such K is referred to as a Lipschitz constant for the function f and f may also be referred to as K-Lipschitz. The smallest constant is sometimes called the (best) Lipschitz constant of f or the dilation or dilatation of f. If K = 1 the function is called a short map, and if 0 ≤ K < 1 and f maps a metric space to itself, the function is called a contraction.
In particular, a real-valued function f : R → R is called Lipschitz continuous if there exists a positive real constant K such that, for all real x1 and x2,
In this case, Y is the set of real numbers R with the standard metric dY(y1, y2) = |y1 − y2|, and X is a subset of R.
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