Editing Lipschitz continuity

{{short description|Strong form of uniform continuity}}
[[File:Lipschitz Visualisierung.gif|thumb|right|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 [[Germany|German]] [[mathematician]] [[Rudolf Lipschitz]], is a strong form of [[uniform continuity]] for [[function (mathematics)|function]]s. 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 continuity|modulus of uniform continuity]]''). For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous.<ref>{{cite book |url=https://books.google.com/books?id=gBPI_oYZoMMC&pg=PA142 |last=Sohrab |first=H. H. |year=2003 |title=Basic Real Analysis |volume=231 |publisher=Birkhäuser |page=142 |isbn=0-8176-4211-0 }}</ref>

In the theory of [[differential equation]]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]]. A special type of Lipschitz continuity, called [[contraction mapping|contraction]], is used in the [[Banach fixed-point theorem]].<ref>{{cite book |first1=Brian S. |last1=Thomson |first2=Judith B. |last2=Bruckner |first3=Andrew M. |last3=Bruckner |title=Elementary Real Analysis |publisher=Prentice-Hall |year=2001 |page=623 |isbn=978-0-13-019075-8 |url=https://books.google.com/books?id=6l_E9OTFaK0C&pg=PA623 }}</ref>

We have the following chain of strict inclusions for functions over a [[Compactness|closed and bounded]] non-trivial interval of the real line:

: '''[[Continuously differentiable]]''' &sub; '''Lipschitz continuous''' &sub; <math>\alpha</math>'''-[[Hölder continuous]]''',

where <math>0 < \alpha \leq 1</math>. We also have

: '''Lipschitz continuous''' &sub; '''[[absolutely continuous]]''' &sub; '''[[uniformly continuous]]''' &sub; '''[[Continuous function|continuous]]'''.

== Definitions ==
Given two [[metric space]]s (''X'', ''d''<sub>''X''</sub>) and (''Y'', ''d''<sub>''Y''</sub>), where ''d''<sub>''X''</sub> denotes the [[metric (mathematics)|metric]] on the set ''X'' and ''d''<sub>''Y''</sub> 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''<sub>1</sub> and ''x''<sub>2</sub> in ''X'',
:<math> d_Y(f(x_1), f(x_2)) \le K d_X(x_1, x_2).</math><ref>{{Citation | last1=Searcóid | first1=Mícheál Ó | title=Metric Spaces |chapter-url=https://books.google.com/books?id=aP37I4QWFRcC&pg=PA154 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer undergraduate mathematics series | isbn=978-1-84628-369-7 | year=2006 |chapter=Lipschitz Functions }}</ref>
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'''<ref>{{cite book |last1=Benyamini |first1=Yoav |last2=Lindenstrauss |first2=Joram |title=Geometric Nonlinear Functional Analysis |date=2000 |publisher=American Mathematical Society |isbn=0-8218-0835-4 |page=11}}</ref> of ''f'' or the '''dilation''' or '''dilatation'''<ref>{{cite book |last1=Burago |first1=Dmitri |last2=Burago |first2=Yuri |last3=Ivanov |first3=Sergei |title=A Course in Metric Geometry |date=2001 |publisher=American Mathematical Society |isbn=0-8218-2129-6}}</ref>{{rp|at=p. 9, Definition 1.4.1}}<ref>{{cite journal |last1=Mahroo |first1=Omar A |last2=Shalchi |first2=Zaid |last3=Hammond |first3=Christopher J |title='Dilatation' and 'dilation': trends in use on both sides of the Atlantic |journal=British Journal of Ophthalmology |date=2014 |volume=98 |issue=6 |pages=845–846 |doi=10.1136/bjophthalmol-2014-304986 |pmid=24568871 |url=https://bjo.bmj.com/content/98/6/845|url-access=subscription }}</ref><ref>{{cite book |last1=Gromov |first1=Mikhael |author1-link=Mikhael Gromov (mathematician) |editor1-last=Rossi |editor1-first=Hugo |title=Prospects in Mathematics: Invited Talks on the Occasion of the 250th Anniversary of Princeton University, March 17-21, 1996, Princeton University |chapter=Quantitative Homotopy Theory |date=1999 |publisher=American Mathematical Society |isbn=0-8218-0975-X |page=46}}</ref> 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 mapping|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 ''x''<sub>1</sub> and ''x''<sub>2</sub>,
:<math> |f(x_1) - f(x_2)| \le K |x_1 - x_2|.</math>
In this case, ''Y'' is the set of [[real number]]s '''R''' with the standard metric ''d''<sub>''Y''</sub>(''y<sub>1</sub>'', ''y<sub>2</sub>'') = |''y<sub>1</sub>'' − ''y<sub>2</sub>''|, and ''X'' is a subset of '''R'''.

In general, the inequality is (trivially) satisfied if ''x''<sub>1</sub> = ''x''<sub>2</sub>. Otherwise, one can equivalently define a function to be Lipschitz continuous [[if and only if]] there exists a constant ''K'' ≥ 0 such that, for all ''x''<sub>1</sub> ≠ ''x''<sub>2</sub>,  
:<math>\frac{d_Y(f(x_1),f(x_2))}{d_X(x_1,x_2)}\le K.</math>
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 [[neighborhood (mathematics)|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 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'' &ge; 0 such that
:<math>d_Y(f(x), f(y)) \leq M d_X(x,  y)^{\alpha}</math> 
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.

{{anchor|Bilipschitz function|Bilipschitz map}}For a real number ''K'' &ge; 1, if
:<math>\frac{1}{K}d_X(x_1,x_2) \le d_Y(f(x_1), f(x_2)) \le K d_X(x_1, x_2)\quad\text{ for all }x_1,x_2\in X,</math>
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 function|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.

==Examples==
;Lipschitz continuous functions that are everywhere differentiable:{{unordered list
 | The function <math>f(x)=\sqrt{x^2+5}</math> defined for all real numbers is Lipschitz continuous with the Lipschitz constant ''K''&nbsp;{{=}}&nbsp;1, because it is everywhere [[Differentiable function|differentiable]] and the absolute value of the derivative is bounded above by 1. See the first property listed below under "[[Lipschitz continuity#Properties|Properties]]".
 | Likewise, the [[sine]] 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:{{unordered list
 |The function <math>f(x) = |x|</math> defined on the reals is Lipschitz continuous with the Lipschitz constant equal to 1, by the [[reverse triangle inequality]]. More generally, a [[norm (mathematics)|norm]] 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:{{unordered list
 | The function <math>f(x) \;=\; \begin{cases} x^2\sin (1/x) & \text{if }x \ne 0 \\ 0 & \text{if }x=0\end{cases}</math>, whose derivative exists but has an essential discontinuity at <math>x=0</math>.
}}
;Continuous functions that are not (globally) Lipschitz continuous:{{unordered list
 | The function ''f''(''x'')&nbsp;{{=}}&nbsp;{{radic|''x''}} defined on [0,&nbsp;1] is ''not'' Lipschitz continuous. This function becomes infinitely steep as ''x'' approaches 0 since its derivative becomes infinite. However, it is uniformly continuous,<ref>{{Citation | last1=Robbin | first1=Joel W. | title=Continuity and Uniform Continuity | url=http://www.math.wisc.edu/~robbin/521dir/cont.pdf}}</ref> and both [[Hölder continuity|Hölder continuous]] of class ''C''<sup>0,  α</sup> for α&nbsp;≤&nbsp;1/2 and also [[absolutely continuous]] on [0,&nbsp;1] (both of which imply the former).
}}
;Differentiable functions that are not (locally) Lipschitz continuous:{{unordered list
 | The function ''f'' defined by ''f''(0)&nbsp;{{=}}&nbsp;0 and ''f''(''x'')&nbsp;{{=}}&nbsp;''x''<sup>3/2</sup>sin(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:{{unordered list
 | The [[exponential function]] becomes arbitrarily steep as ''x'' → ∞, and therefore is ''not'' globally Lipschitz continuous, despite being an [[analytic function]].
 | The function ''f''(''x'')&nbsp;{{=}}&nbsp;''x''<sup>2</sup> with domain all real numbers is ''not'' Lipschitz continuous. This function becomes arbitrarily steep as ''x'' approaches infinity. It is however locally Lipschitz continuous.
}}

==Properties==
*An everywhere differentiable function ''g''&nbsp;:&nbsp;'''R'''&nbsp;→&nbsp;'''R''' is Lipschitz continuous (with ''K''&nbsp;=&nbsp;sup&nbsp;|''g''′(''x'')|) if and only if it has a bounded [[first derivative]]; one direction follows from the [[mean 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 function ''g''&nbsp;:&nbsp;'''R'''&nbsp;→&nbsp;'''R''' is [[absolutely continuous]] and therefore is differentiable [[almost everywhere]], that is, differentiable at every point outside a set of [[Lebesgue measure]] zero.  Its derivative is [[essentially bounded]] in magnitude by the Lipschitz constant, and for ''a''&nbsp;< ''b'', the difference ''g''(''b'')&nbsp;−&nbsp;''g''(''a'') is equal to the integral of the derivative ''g''′ on the interval [''a'',&nbsp;''b''].
**Conversely, if ''f''&nbsp;: ''I''&nbsp;→ '''R''' is absolutely continuous and thus differentiable almost everywhere, and satisfies |''f′''(''x'')|&nbsp;≤ ''K'' for almost all ''x'' in ''I'', then ''f'' is Lipschitz continuous with Lipschitz constant at most ''K''.
**More generally, [[Rademacher's theorem]] extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ''f''&nbsp;:&nbsp;''U''&nbsp;→&nbsp;'''R'''<sup>''m''</sup>, where ''U'' is an open set in '''R'''<sup>''n''</sup>, is [[almost everywhere]] [[derivative|differentiable]]. Moreover, if ''K'' is the best Lipschitz constant of ''f'', then <math>\|Df(x)\|\le K</math> whenever the [[total derivative]] ''Df'' exists.{{citation needed|date=March 2023}}
*For a differentiable Lipschitz map <math>f: U \to \R^m</math> the inequality <math>\|Df\|_{W^{1,\infty}(U)}\le K</math> holds for the best Lipschitz constant <math>K</math> of <math>f</math>. If the domain <math>U</math> is convex then in fact <math>\|Df\|_{W^{1,\infty}(U)}= K</math>.{{Explain|date=November 2019}}
*Suppose that {''f<sub>n</sub>''} is a sequence of Lipschitz continuous mappings between two metric spaces, and that all ''f<sub>n</sub>'' have Lipschitz constant bounded by some ''K''.  If ''f<sub>n</sub>'' converges to a mapping ''f'' [[uniform convergence|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 ''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&ndash;Weierstrass theorem]] (or as a consequence of [[Weierstrass approximation theorem]], because every polynomial is locally Lipschitz continuous).
*Every Lipschitz continuous map is [[uniformly continuous]], and hence [[continuous function|continuous]]. More generally, a set of functions with bounded Lipschitz constant forms an  [[equicontinuous]] set.  The [[Arzelà–Ascoli theorem]] implies that if {''f<sub>n</sub>''} is a [[uniformly 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 space ''X'' having Lipschitz constant ≤&nbsp;''K''&thinsp; is a [[Locally compact space|locally compact]] convex subset of the Banach space ''C''(''X'').
*For a family of Lipschitz continuous functions ''f''<sub>α</sub> with common constant, the function <math>\sup_\alpha f_\alpha</math> (and <math>\inf_\alpha f_\alpha</math>) 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''&nbsp;: ''U''&nbsp;→ '''R''' is a Lipschitz continuous function, there always exist Lipschitz continuous maps ''M''&nbsp;→ '''R''' that extend ''f'' and have the same Lipschitz constant as ''f'' (see also [[Kirszbraun theorem]]). An extension is provided by 
::<math>\tilde f(x):=\inf_{u\in U}\{ f(u)+k\, d(x,u)\},</math> 
:where ''k'' is  a Lipschitz constant for ''f'' on ''U''.

==Lipschitz manifolds==
A '''Lipschitz structure''' on a [[topological manifold]] is defined using an [[atlas (topology)|atlas of charts]] whose transition maps are bilipschitz; this is possible because bilipschitz maps form a [[pseudogroup]].  Such a structure allows one to define locally Lipschitz maps between such manifolds, similarly to how one defines smooth maps between [[smooth manifold]]s: if {{mvar|M}} and {{mvar|N}} are Lipschitz manifolds, then a function <math>f:M \to N</math> is '''locally Lipschitz''' if and only if for every pair of coordinate charts <math>\phi:U \to M</math> and <math>\psi:V \to N</math>, where {{mvar|U}} and {{mvar|V}} are open sets in the corresponding Euclidean spaces, the composition
<math display="block">\psi^{-1} \circ f \circ \phi:U \cap (f \circ \phi)^{-1}(\psi(V)) \to V</math>
is locally Lipschitz.  This definition does not rely on defining a metric on {{mvar|M}} or {{mvar|N}}.<ref name="Rosenberg">{{cite conference |first=Jonathan |last=Rosenberg |author-link=Jonathan Rosenberg (mathematician) |book-title=Miniconferences on harmonic analysis and operator algebras (Canberra, 1987) |title=Applications of analysis on Lipschitz manifolds |year=1988 |publisher=[[Australian National University]] |location=Canberra |pages=269–283 |url=https://projecteuclid.org/proceedings/proceedings-of-the-centre-for-mathematics-and-its-applications/Miniconference-on-Harmonic-Analysis-and-Operator-Algebras/Chapter/Applications-of-analysis-on-Lipschitz-manifolds/pcma/1416336222}} {{MathSciNet|id=954004}}</ref>

This structure is intermediate between that of a [[piecewise-linear manifold]] and a [[topological manifold]]: a PL structure gives rise to a unique Lipschitz structure.<ref>{{SpringerEOM|title=Topology of manifolds}}</ref>  While Lipschitz manifolds are closely related to topological manifolds, [[Rademacher's theorem]] allows one to do analysis, yielding various applications.<ref name="Rosenberg"/>

==One-sided Lipschitz==
Let ''F''(''x'') be an [[Semi-continuity|upper semi-continuous]] function of ''x'', and that ''F''(''x'') is a closed, convex set for all ''x''. Then ''F'' is one-sided Lipschitz<ref>{{cite journal |last1=Donchev |first1=Tzanko |last2=Farkhi |first2=Elza |year=1998 |title=Stability and Euler Approximation of One-sided Lipschitz Differential Inclusions |journal=SIAM Journal on Control and Optimization |volume=36 |issue=2 |pages=780–796 |doi=10.1137/S0363012995293694 }}</ref> if
:<math>(x_1-x_2)^T(F(x_1)-F(x_2))\leq C\Vert x_1-x_2\Vert^2</math> 
for some ''C'' and for all ''x''<sub>1</sub> and ''x''<sub>2</sub>.

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

:<math>\begin{cases}
F:\mathbf{R}^2\to\mathbf{R},\\ 
F(x,y)=-50(y-\cos(x))
\end{cases}</math>

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''<sup>−''x''</sup>, with ''C'' = 0.

==See also==

* {{annotated link|Contraction mapping}}
* [[Dini continuity]]
* [[Modulus of continuity]]
* [[Quasi-isometry]]
* [[Johnson–Lindenstrauss lemma]] – For any integer ''n''≥0, any finite subset ''X''⊆'''''R'''<sup>n</sup>'', and any real number 0<ε<1, there exists a (1+ε)-bi-Lipschitz function <math>f:\mathbb R^n\to\mathbb R^d,</math> where <math>d=\lceil15(\ln|X|)/\varepsilon^2\rceil.</math>

==References==

{{reflist}}

{{Metric spaces}}

[[Category:Lipschitz maps| ]]
[[Category:Structures on manifolds]]