In

_{''X''}) and (''Y'', ''d''_{''Y''}), where ''d''_{''X''} denotes the _{''Y''} 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 ''x''_{1} and ''x''_{2} in ''X'',
:$d\_Y(f(x\_1),\; f(x\_2))\; \backslash le\; K\; d\_X(x\_1,\; x\_2).$
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 _{1} and ''x''_{2},
:$,\; f(x\_1)\; -\; f(x\_2),\; \backslash le\; K\; ,\; x\_1\; -\; x\_2,\; .$
In this case, ''Y'' is the set of _{''Y''}(''y_{1}'', ''y_{2}'') = , ''y_{1}'' − ''y_{2}'', , and ''X'' is a subset of R.
In general, the inequality is (trivially) satisfied if ''x''_{1} = ''x''_{2}. Otherwise, one can equivalently define a function to be Lipschitz continuous _{1} ≠ ''x''_{2},
:$\backslash frac\backslash le\; K.$
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 outside of this cone (see figure).
A function is called locally Lipschitz continuous if for every ''x'' in ''X'' there exists a

^{''m''}, where ''U'' is an open set in R^{''n''}, is _{n}'' have Lipschitz constant bounded by some ''K''. If ''f_{n}'' 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 _{α} with common constant, the function $\backslash sup\_\backslash alpha\; f\_\backslash alpha$ (and $\backslash inf\_\backslash alpha\; f\_\backslash alpha$) is Lipschitz continuous as well, with the same Lipschitz constant, provided it assumes a finite 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
::$\backslash tilde\; f(x):=\backslash inf\_\backslash ,$
:where ''k'' is a Lipschitz constant for ''f'' on ''U''.

_{1} and ''x''_{2}.
It is possible that the function ''F'' could have a very large Lipschitz constant but a moderately sized, or even negative, one-sided Lipschitz constant. For example, the function
:$\backslash begin\; F:\backslash mathbf^2\backslash to\backslash mathbf,\backslash \backslash \; F(x,y)=-50(y-\backslash cos(x))\; \backslash end$
has Lipschitz constant ''K'' = 50 and 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.

^{n}'', and any real number 0<ε<1, there exists a (1+ε)-bi-Lipschitz function $f:\backslash mathbb\; R^n\backslash to\backslash mathbb\; R^d,$ where $d=\backslash lceil15(\backslash ln,\; X,\; )/\backslash varepsilon^2\backslash rceil.$

mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, Lipschitz continuity, named after German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Rudolf Lipschitz, is a strong form of uniform continuity
In mathematics, a real function (mathematics), function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to ea ...

for functions. Intuitively, a Lipschitz continuous function
In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value ...

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
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign (mathematics), sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative number, negative (in which cas ...

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 (or '' modulus of uniform continuity''). For instance, every function that has bounded first derivatives is Lipschitz continuous.
In the theory of differential equation
In mathematics, a differential equation is an functional equation, equation that relates one or more unknown function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the der ...

s, 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
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function (mathematics), function at a given point in the domain of a functio ...

. A special type of Lipschitz continuity, called contraction
Contraction may refer to:
Linguistics
* Contraction (grammar), a shortened word
* Poetic contraction, omission of letters for poetic reasons
* Elision, omission of sounds
** Syncope (phonology), omission of sounds in a word
* Synalepha, merged ...

, is used in the Banach fixed-point theorem
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...

.
We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line:
: Continuously differentiable
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

⊂ Lipschitz continuous ⊂ $\backslash alpha$- Hölder continuous,
where $0\; <\; \backslash alpha\; \backslash leq\; 1$. We also have
: Lipschitz continuous ⊂ absolutely continuous
In calculus, absolute continuity is a smoothness (mathematics), smoothness property of function (mathematics), functions that is stronger than continuous function, continuity and uniform continuity. The notion of absolute continuity allows one to o ...

⊂ uniformly continuous
In mathematics, a real function (mathematics), function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to ea ...

.
Definitions

Given twometric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...

s (''X'', ''d''metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...

on the set ''X'' and ''d''short map In the mathematics, mathematical theory of metric spaces, a metric map is a Function (mathematics), function between metric spaces that does not increase any distance (such functions are always continuous function, continuous).
These maps are the m ...

, and if 0 ≤ ''K'' < 1 and ''f'' maps a metric space to itself, the function is called a contraction
Contraction may refer to:
Linguistics
* Contraction (grammar), a shortened word
* Poetic contraction, omission of letters for poetic reasons
* Elision, omission of sounds
** Syncope (phonology), omission of sounds in a word
* Synalepha, merged ...

.
In particular, a real-valued function
In mathematics, a real-valued function is a function whose values are real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, durati ...

''f'' : ''R'' → ''R'' is called Lipschitz continuous if there exists a positive real constant K such that, for all real ''x''real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s R with the standard metric ''d''if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...

there exists a constant ''K'' ≥ 0 such that, for all ''x''neighborhood
A neighbourhood (British English
British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...

''U'' of ''x'' such that ''f'' restricted to ''U'' is Lipschitz continuous. Equivalently, if ''X'' is a locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...

metric space, then ''f'' is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of ''X''. In spaces that are not locally compact, this is a necessary but not a sufficient condition.
More generally, a function ''f'' defined on ''X'' is said to be Hölder continuous or to satisfy a Hölder condition of order α > 0 on ''X'' if there exists a constant ''M'' ≥ 0 such that
:$d\_Y(f(x),\; f(y))\; \backslash leq\; M\; d\_X(x,\; y)^$
for all ''x'' and ''y'' in ''X''. Sometimes a Hölder condition of order α is also called a uniform Lipschitz condition of order α > 0.
For a real number ''K'' ≥ 1, if
:$\backslash fracd\_X(x\_1,x\_2)\; \backslash le\; d\_Y(f(x\_1),\; f(x\_2))\; \backslash le\; K\; d\_X(x\_1,\; x\_2)\backslash quad\backslash textx\_1,x\_2\backslash in\; X,$
then ''f'' is called ''K''-bilipschitz (also written ''K''-bi-Lipschitz). We say ''f'' is bilipschitz or bi-Lipschitz to mean there exists such a ''K''. A bilipschitz mapping is injective
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

, and is in fact a homeomorphism
In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...

onto its image. A bilipschitz function is the same thing as an injective Lipschitz function whose inverse function
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

is also Lipschitz.
Examples

;Lipschitz continuous functions: ;Lipschitz continuous functions that are not everywhere differentiable: ;Lipschitz continuous functions that are everywhere differentiable but not continuously differentiable: ;Continuous functions that are not (globally) Lipschitz continuous: ;Differentiable functions that are not (locally) Lipschitz continuous: ;Analytic functions that are not (globally) Lipschitz continuous:Properties

*An everywhere differentiable function ''g'' : R → R is Lipschitz continuous (with ''K'' = sup , ''g''′(''x''), ) if and only if it has boundedfirst derivative
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...

; one direction follows from the mean value theorem
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

. 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 function ''g'' : R → R is absolutely continuous
In calculus, absolute continuity is a smoothness (mathematics), smoothness property of function (mathematics), functions that is stronger than continuous function, continuity and uniform continuity. The notion of absolute continuity allows one to o ...

and therefore is differentiable almost everywhere
In measure theory (a branch of mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, meas ...

, that is, differentiable at every point outside a set of Lebesgue measure
In Measure (mathematics), measure theory, a branch of mathematics, the Lebesgue measure, named after france, French mathematician Henri Lebesgue, is the standard way of assigning a measure (mathematics), measure to subsets of ''n''-dimensional Eucl ...

zero. Its derivative is essentially bounded
Essence ( la, essentia) is a polysemic term, used in philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such ...

in magnitude by the Lipschitz constant, and for ''a'' < ''b'', the difference ''g''(''b'') − ''g''(''a'') is equal to the integral of the derivative ''g''′ on the interval 'a'', ''b''
**Conversely, if ''f'' : ''I'' → R is absolutely continuous and thus differentiable almost everywhere, and satisfies , ''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 In mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infi ...

extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ''f'' : ''U'' → Ralmost everywhere
In measure theory (a branch of mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, meas ...

differentiable
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

. Moreover, if ''K'' is the best Lipschitz constant of ''f'', then $\backslash ,\; Df(x)\backslash ,\; \backslash le\; K$ whenever the total derivative
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

''Df'' exists.
*For a differentiable Lipschitz map $f:\; U\; \backslash to\; \backslash R^m$ the inequality $\backslash ,\; Df\backslash ,\; \_\backslash le\; K$ holds for the best Lipschitz constant $K$ of $f$. If the domain $U$ is convex then in fact $\backslash ,\; Df\backslash ,\; \_=\; K$.
*Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all ''fBanach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a Complete metric space, complete normed vector space. Thus, a Banach space is a vector space with a Metric (mathematics), metric that allows the computation ...

of continuous functions. This result does not hold for sequences in which the functions may have ''unbounded'' 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 the Stone–Weierstrass theorem
In mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, inf ...

(or as a consequence of Weierstrass approximation theorem
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve ma ...

, because every polynomial is locally Lipschitz continuous).
*Every Lipschitz continuous map is uniformly continuous
In mathematics, a real function (mathematics), function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to ea ...

, and hence '' a fortiori'' continuous. More generally, a set of functions with bounded Lipschitz constant forms an equicontinuous
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous function, continuous and they have equal variation over a given neighbourhood (mathematics), neighbourhood, in a precise sense described herein.
...

set. The Arzelà–Ascoli theorem
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence (mathematics), sequence of a given family of real number, real-valued continuous functions def ...

implies that if is a uniformly bounded
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...

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 space ''X'' having Lipschitz constant ≤ ''K'' is a locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...

convex subset of the Banach space ''C''(''X'').
*For a family of Lipschitz continuous functions ''f''Lipschitz manifolds

A Lipschitz structure on atopological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological spa ...

is defined using an atlas of charts whose transition maps are bilipschitz; this is possible because bilipschitz maps form a pseudogroup In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group (mathematics), group, originating however from the geometric ap ...

. Such a structure allows one to define locally Lipschitz maps between such manifolds, similarly to how one defines smooth maps between smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spa ...

s: if and are Lipschitz manifolds, then a function $f:M\; \backslash to\; N$ is locally Lipschitz if and only if for every pair of coordinate charts $\backslash phi:U\; \backslash to\; M$ and $\backslash psi:V\; \backslash to\; N$, where and are open sets in the corresponding Euclidean spaces, the composition
$$\backslash psi^\; \backslash circ\; f\; \backslash circ\; \backslash phi:U\; \backslash cap\; (f\; \backslash circ\; \backslash phi)^(\backslash psi(V))\; \backslash to\; N$$
is locally Lipschitz. This definition does not rely on defining a metric on or .
This structure is intermediate between that of a piecewise-linear manifold and a topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological spa ...

: a PL structure gives rise to a unique Lipschitz structure. While Lipschitz manifolds are closely related to topological manifolds, Rademacher's theorem In mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infi ...

allows one to do analysis, yielding various applications.
One-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 if :$(x\_1-x\_2)^T(F(x\_1)-F(x\_2))\backslash leq\; C\backslash Vert\; x\_1-x\_2\backslash Vert^2$ for some ''C'' and for all ''x''See also

* * Dini continuity * Modulus of continuity * Quasi-isometry * Johnson-Lindenstrauss lemma – For any integer ''n''≥0, any finite subset ''X''⊆''RReferences

{{reflist Structures on manifolds