Editing Lipschitz continuity (section)

{{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]]'''.