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
Inmathematical analysis,Lipschitz continuity, named afterGermanmathematicianRudolf Lipschitz, is a strong form ofuniform continuity forfunctions. Intuitively, a Lipschitzcontinuous 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, theabsolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called theLipschitz constant of the function (and is related to themodulus of uniform continuity). For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous.[1]
Given twometric spaces (X,dX) and (Y,dY), wheredX denotes themetric on the setX anddY is the metric on setY, a functionf :X →Y is calledLipschitz continuous if there exists a real constantK ≥ 0 such that, for allx1 andx2 inX,
Any suchK is referred to as aLipschitz constant for the functionf, andf may also be referred to asK-Lipschitz. The smallest constant is sometimes calledthe (best) Lipschitz constant[4] off or thedilation[5][6] off. IfK = 1 the function is called ashort map, and if 0 ≤K < 1 andf maps a metric space to itself, the function is called acontraction.
In particular, areal-valued functionf :R →R is called Lipschitz continuous if there exists a positive real constant K such that, for all realx1 andx2,
In this case,Y is the set ofreal numbersR with the standard metricdY(y1,y2) = |y1 −y2|, andX is a subset ofR.
In general, the inequality is (trivially) satisfied ifx1 =x2. Otherwise, one can equivalently define a function to be Lipschitz continuousif and only if there exists a constantK ≥ 0 such that, for allx1 ≠x2,
For real-valued functions of several real variables, this holds if and only if the absolute value of the slopes of all secant lines are bounded byK. The set of lines of slopeK passing through a point on the graph of the function forms a circular cone, and a function is Lipschitz if and only if the graph of the function everywhere lies completely outside of this cone (see figure).
A function is calledlocally Lipschitz continuous if for everyx inX there exists aneighborhoodU ofx such thatf restricted toU is Lipschitz continuous. Equivalently, ifX is alocally compact metric space, thenf is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset ofX. In spaces that are not locally compact, this is a necessary but not a sufficient condition.
More generally, a functionf defined onX is said to beHölder continuous or to satisfy aHölder condition of order α > 0 onX if there exists a constantM ≥ 0 such that
for allx andy inX. Sometimes a Hölder condition of order α is also called auniform Lipschitz condition of order α > 0.
For a real numberK ≥ 1, if
thenf is calledK-bilipschitz (also writtenK-bi-Lipschitz). We sayf isbilipschitz orbi-Lipschitz to mean there exists such aK. A bilipschitz mapping isinjective, and is in fact ahomeomorphism onto its image. A bilipschitz function is the same thing as an injective Lipschitz function whoseinverse function is also Lipschitz.
Lipschitz continuous functions that are everywhere differentiable
The function defined for all real numbers is Lipschitz continuous with the Lipschitz constantK = 1, because it is everywheredifferentiable and the absolute value of the derivative is bounded above by 1. See the first property listed below under "Properties".
Likewise, thesine function is Lipschitz continuous because its derivative, the cosine function, is bounded above by 1 in absolute value.
Lipschitz continuous functions that are not everywhere differentiable
The function defined on the reals is Lipschitz continuous with the Lipschitz constant equal to 1, by thereverse triangle inequality. More generally, anorm on a vector space is Lipschitz continuous with respect to the associated metric, with the Lipschitz constant equal to 1.
Lipschitz continuous functions that are everywhere differentiable but not continuously differentiable
The function, whose derivative exists but has an essential discontinuity at.
Continuous functions that are not (globally) Lipschitz continuous
The functionf(x) = √x defined on [0, 1] isnot Lipschitz continuous. This function becomes infinitely steep asx approaches 0 since its derivative becomes infinite. However, it is uniformly continuous,[7] and bothHölder continuous of classC0, α for α ≤ 1/2 and alsoabsolutely continuous on [0, 1] (both of which imply the former).
Differentiable functions that are not (locally) Lipschitz continuous
The functionf defined byf(0) = 0 andf(x) = x3/2sin(1/x) for 0<x≤1 gives an example of a function that is differentiable on a compact set while not locally Lipschitz because its derivative function is not bounded. See also the first property below.
Analytic functions that are not (globally) Lipschitz continuous
The functionf(x) = x2 with domain all real numbers isnot Lipschitz continuous. This function becomes arbitrarily steep asx approaches infinity. It is however locally Lipschitz continuous.
An everywhere differentiable functiong : R → R is Lipschitz continuous (withK = sup |g′(x)|) if and only if it has a boundedfirst derivative; one direction follows from themean value theorem. In particular, any continuously differentiable function is locally Lipschitz, as continuous functions are locally bounded so its gradient is locally bounded as well.
A Lipschitz functiong : R → R isabsolutely continuous and therefore is differentiablealmost everywhere, that is, differentiable at every point outside a set ofLebesgue measure zero. Its derivative isessentially bounded in magnitude by the Lipschitz constant, and fora <b, the differenceg(b) − g(a) is equal to the integral of the derivativeg′ on the interval [a, b].
Conversely, iff :I →R is absolutely continuous and thus differentiable almost everywhere, and satisfies |f′(x)| ≤K for almost allx inI, thenf is Lipschitz continuous with Lipschitz constant at mostK.
For a differentiable Lipschitz map the inequality holds for the best Lipschitz constant of. If the domain is convex then in fact.[further explanation needed]
Suppose that {fn} is a sequence of Lipschitz continuous mappings between two metric spaces, and that allfn have Lipschitz constant bounded by someK. Iffn converges to a mappingfuniformly, thenf is also Lipschitz, with Lipschitz constant bounded by the sameK. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of theBanach space of continuous functions. This result does not hold for sequences in which the functions may haveunbounded Lipschitz constants, however. In fact, the space of all Lipschitz functions on a compact metric space is a subalgebra of the Banach space of continuous functions, and thus dense in it, an elementary consequence of theStone–Weierstrass theorem (or as a consequence ofWeierstrass approximation theorem, because every polynomial is locally Lipschitz continuous).
Every Lipschitz continuous map isuniformly continuous, and hencecontinuous. More generally, a set of functions with bounded Lipschitz constant forms anequicontinuous set. TheArzelà–Ascoli theorem implies that if {fn} is auniformly bounded sequence of functions with bounded Lipschitz constant, then it has a convergent subsequence. By the result of the previous paragraph, the limit function is also Lipschitz, with the same bound for the Lipschitz constant. In particular the set of all real-valued Lipschitz functions on a compact metric spaceX having Lipschitz constant ≤ K is alocally compact convex subset of the Banach spaceC(X).
For a family of Lipschitz continuous functionsfα with common constant, the function (and) is Lipschitz continuous as well, with the same Lipschitz constant, provided it assumes a finite value at least at a point.
IfU is a subset of the metric spaceM andf :U →R is a Lipschitz continuous function, there always exist Lipschitz continuous mapsM →R that extendf and have the same Lipschitz constant asf (see alsoKirszbraun theorem). An extension is provided by
ALipschitz structure on atopological manifold is defined using anatlas of charts whose transition maps are bilipschitz; this is possible because bilipschitz maps form apseudogroup. Such a structure allows one to define locally Lipschitz maps between such manifolds, similarly to how one defines smooth maps betweensmooth manifolds: ifM andN are Lipschitz manifolds, then a function islocally Lipschitz if and only if for every pair of coordinate charts and, whereU andV are open sets in the corresponding Euclidean spaces, the compositionis locally Lipschitz. This definition does not rely on defining a metric onM orN.[8]
This structure is intermediate between that of apiecewise-linear manifold and atopological manifold: a PL structure gives rise to a unique Lipschitz structure.[9] While Lipschitz manifolds are closely related to topological manifolds,Rademacher's theorem allows one to do analysis, yielding various applications.[8]
LetF(x) be anupper semi-continuous function ofx, and thatF(x) is a closed, convex set for allx. ThenF is one-sided Lipschitz[10] if
for someC and for allx1 andx2.
It is possible that the functionF could have a very large Lipschitz constant but a moderately sized, or even negative, one-sided Lipschitz constant. For example, the function
has Lipschitz constantK = 50 and a one-sided Lipschitz constantC = 0. An example which is one-sided Lipschitz but not Lipschitz continuous isF(x) =e−x, withC = 0.
^Benyamini, Yoav; Lindenstrauss, Joram (2000).Geometric Nonlinear Functional Analysis. American Mathematical Society. p. 11.ISBN0-8218-0835-4.
^Burago, Dmitri; Burago, Yuri; Ivanov, Sergei (2001).A Course in Metric Geometry. American Mathematical Society.ISBN0-8218-2129-6.
^Gromov, Mikhael (1999). "Quantitative Homotopy Theory". In Rossi, Hugo (ed.).Prospects in Mathematics: Invited Talks on the Occasion of the 250th Anniversary of Princeton University, March 17-21, 1996, Princeton University. American Mathematical Society. p. 46.ISBN0-8218-0975-X.
^Donchev, Tzanko; Farkhi, Elza (1998). "Stability and Euler Approximation of One-sided Lipschitz Differential Inclusions".SIAM Journal on Control and Optimization.36 (2):780–796.doi:10.1137/S0363012995293694.
