Lipschitz continuity

In mathematical analysis,Lipschitz continuity,named after Rudolf Lipschitz,is a strong form of uniform con-tinuity for functions.Intuitively,a Lipschitz continuous function is limited in how fast it can change:there exists a de?nite 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;this bound is called a Lipschitz constant of the function(or modulus of uniform continuity).For in-stance,every function that has bounded?rst derivatives is Lipschitz.[1]

In the theory of di?erential equations,Lipschitz continu-ity is the central condition of the Picard–Lindel?f the-orem 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?xed point theorem.

We have the following chain of inclusions for functions over a closed and bounded[2]subset of the real line Continuously di?erentiable?Lipschitz

continuous?α-H?lder continuous?

uniformly continuous=continuous

where0<α≤1.We also have

Lipschitz continuous?absolutely contin-

uous?bounded variation?di?erentiable

almost everywhere

1De?nitions

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(for example,Y might be the set of real numbers R with the metric dY(y1,y2)=|y1?y2|,and X might be a subset of R),a function f:X→Y is called Lipschitz continuous if there exists a real constant K≥0such that, for all x1and x2in X,

d Y(f(x1),f(x2))≤Kd X(x1,x2).[3]

Any such K is referred to as a Lipschitz constant for the function f.The smallest constant is sometimes called the(best)Lipschitz constant;however,in most cases, the latter notion is less relevant.If K=1the function is called a short map,and if0≤K<1the function is called a contraction

For a Lipschitz continuous function,there is a double cone (shown in white)whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone.

The inequality is(trivially)satis?ed if x1=x2.Otherwise, one can equivalently de?ne a function to be Lipschitz con-tinuous if and only if there exists a constant K≥0such that,for all x1≠x2,

d Y(f(x1),f(x2))

d X(x1,x2)

≤K.

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 by K.The set of lines of slope K 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 out-side of this cone(see?gure).

A function is called locally Lipschitz continuous if for every x in X there exists a neighborhood U of x such that f restricted to U is Lipschitz continuous.Equivalently,if X is a locally compact metric space,then f is locally Lip-schitz if and only if it is Lipschitz continuous on every compact subset of X.In spaces that are not locally com-pact,this is a necessary but not a su?cient condition. More generally,a function f de?ned on X is said to be H?lder continuous or to satisfy a H?lder condition of orderα>0on X if there exists a constant M>0such that

d Y(f(x),f(y))≤Md X(x,y)α

for all x and y in X.Sometimes a H?lder condition of orderαis also called a uniform Lipschitz condition of

23PROPERTIES

orderα>0.

If there exists a K≥1with

d X(x1,x2)≤d Y(f(x1),f(x2))≤Kd X(x1,x2)

then f is called bilipschitz(also written bi-Lipschitz).

A bilipschitz mapping is injective,and is in fact a

homeomorphism onto its image.A bilipschitz function is

the same thing as an injective Lipschitz function whose

inverse function is also Lipschitz.

2Examples

Lipschitz continuous functions?The function

f(x)=√x2+5de?ned for all real numbers

is Lipschitz continuous with the Lipschitz

constant K=1,because it is everywhere

di?erentiable and the absolute value of the

derivative is bounded above by1.See the?rst

property listed below under"Properties".

?Likewise,the sine function is Lipschitz con-

tinuous because its derivative,the cosine func-

tion,is bounded above by1in absolute value.

?The function f(x)=|x|de?ned on the reals is

Lipschitz continuous with the Lipschitz con-

stant equal to1,by the reverse triangle inequal-

ity.This is an example of a Lipschitz contin-

uous function that is not di?erentiable.More

generally,a norm on a vector space is Lips-

chitz continuous with respect to the associated

metric,with the Lipschitz constant equal to1.

Lipschitz continuous functions that are not every-where di?erentiable ?

The function f(x)=|x|.

Continuous functions that are not(globally)Lips-chitz continuous

?The function f(x)=√x de?ned on[0,1]is not

Lipschitz continuous.This function becomes

in?nitely steep as x approaches0since its

derivative becomes in?nite.However,it is

uniformly continuous[4]as well as H?lder

continuous of class C0,αforα≤1/2.

Di?erentiable functions that are not(globally)Lips-chitz continuous ?

The function f(x)=x3/2sin(1/x)where x≠0and f(0)=0,restricted on[0,1],gives an example of a function that is di?erentiable on a compact set while not locally Lipschitz because its derivative function is not bounded. See also the?rst property below.

Analytic functions that are not(globally)Lipschitz continuous ?

The exponential function becomes arbitrarily

steep as x→∞,and therefore is not glob-

ally Lipschitz continuous,despite being an

analytic function.

?The function f(x)=x2with domain all real

numbers is not Lipschitz continuous.This

function becomes arbitrarily steep as x ap-

proaches in?nity.It is however locally Lips-

chitz continuous.

3Properties

?An everywhere di?erentiable function g:R→R is Lipschitz continuous(with K=sup|g′(x)|)if and only if it has bounded?rst derivative;one direction follows from the mean value theorem.In particu-lar,any continuously di?erentiable function is lo-cally Lipschitz,as continuous functions are locally bounded so its gradient is locally bounded as well.

?A Lipschitz function g:R→R is absolutely con-tinuous and therefore is di?erentiable almost every-where,that is,di?erentiable at every point outside

a set of Lebesgue measure zero.Its derivative is

essentially bounded in magnitude by the Lipschitz constant,and for a

?Conversely,if f:I→R is absolutely continu-

ous and thus di?erentiable almost everywhere,

and satis?es|f′(x)|≤K for almost all x in I,

then f is Lipschitz continuous with Lipschitz

constant at most K.

?More generally,Rademacher’s theorem ex-

tends the di?erentiability result to Lipschitz

mappings between Euclidean spaces:a Lips-

chitz map f:U→R m,where U is an open

set in R n,is almost everywhere di?erentiable.

Moreover,if K is the best Lipschitz constant

of f,then∥Df(x)∥≤K whenever the total

derivative Df exists.

?For a di?erentiable Lipschitz map f:U→R m the inequality∥Df∥∞,U≤K holds for the best Lips-chitz constant of f,and it turns out to be an equality if the domain U is convex.

?Suppose that{fn}is a sequence of Lipschitz contin-uous mappings between two metric spaces,and that all fn have Lipschitz constant bounded by some K.

If fn converges to a mapping f uniformly,then f is also Lipschitz,with Lipschitz constant bounded by the same K.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 the Banach space of continuous functions.This result does not hold

for sequences in which the functions may have un-bounded Lipschitz constants,however.In fact,the space of all Lipschitz functions on a compact metric space is a subalgebra of the Banach space of contin-uous functions,and thus dense in it,an elementary consequence of the Stone–Weierstrass theorem(or as a consequence of Weierstrass approximation the-orem,because every polynomial is Lipschitz contin-uous).

?Every Lipschitz continuous map is uniformly con-tinuous,and hence a fortiori continuous.More gen-erally,a set of functions with bounded Lipschitz constant forms an equicontinuous set.The Arzelà–Ascoli theorem implies that if{fn}is a uniformly bounded sequence of functions with bounded Lips-chitz 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 space X having Lipschitz constant≤K is a locally compact convex subset of the Banach space C(X).

?For a family of Lipschitz continuous functions fαwith common constant,the function supαfα(and infαfα)is Lipschitz continuous as well,with the same Lipschitz constant,provided it assumes a?nite value at least at a point.

?If U is a subset of the metric space M and f:U→R is a Lipschitz continuous function,there always exist Lipschitz continuous maps M→R which extend f and have the same Lipschitz constant as f(see also Kirszbraun theorem).An extension is provided by

?f(x):=inf

u∈U

{f(u)+k d(x,u)},

where k is a Lipschitz constant for f on U.

4Lipschitz manifolds

Let U and V be two open sets in R n.A function T:U→V is called bi-Lipschitz if it is a Lipschitz homeomorphism onto its image,and its inverse is also Lipschitz.

Using bi-Lipschitz mappings,it is possible to de?ne a Lipschitz structure on a topological manifold,since there is a pseudogroup structure on bi-Lipschitz homeomor-phisms.This structure is intermediate between that of a piecewise-linear manifold and a smooth manifold.In fact a PL structure gives rise to a unique Lipschitz structure;[5] it can in that sense'nearly'be smoothed.5One-sided Lipschitz

Let F(x)be an upper semi-continuous function of x,and that F(x)is a closed,convex set for all x.Then F is one-sided Lipschitz[6]if

(x1?x2)T(F(x1)?F(x2))≤C∥x1?x2∥2

for some C for all x1and x2.

It is possible that the function F could have a very large Lipschitz constant but a moderately sized,or even nega-tive,one-sided Lipschitz constant.For example the func-tion

{

F:R2→R,

F(x,y)=?50(y?cos(x))

has Lipschitz constant K=50and a one-sided Lipschitz constant C=0.An example which is one-sided Lipschitz but not Lipschitz continuous is F(x)=e?x,with C= 0.

6See also

?Dini continuity

?Modulus of continuity

7References

[1]Sohrab,H.H.(2003).Basic real analysis(Vol.231).

Birkh?user

[2]Compactness

[3]Searcóid,Mícheáló(2006),Metric spaces,Springer

undergraduate mathematics series,Berlin,New York: Springer-Verlag,ISBN978-1-84628-369-7,section9.4 [4]Robbin,Joel W.,Continuity and Uniform Continuity

(PDF)

[5]SpringerLink:Topology of manifolds

[6]Donchev,Tzanko;Farkhi,Elza(1998).“Stability and Eu-

ler Approximation of One-sided Lipschitz Di?erential In-clusions”.SIAM Journal on Control and Optimization.36

(2):780–796.doi:10.1137/S0363012995293694.

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