Editing Lipschitz continuity (section)

== 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.