Lipschitz-continuity

LipschitzcontinuityInmathematicalanalysis,Lipschitzcontinuity,namedafterRudolfLipschitz,isastrongformofuniformcon-tinuityforfunctions.Intuitively,aLipschitzcontinuousfunctionislimitedinhowfastitcanchange:thereexistsadefiniterealnumbersuchthat,foreverypairofpointsonthegraphofthisfunction,theabsolutevalueoftheslopeofthelineconnectingthemisnotgreaterthanthisrealnumber;thisboundiscalledaLipschitzconstantofthefunction(ormodulusofuniformcontinuity).Forin-stance,everyfunctionthathasboundedfirstderivativesisLipschitz.[1]Inthetheoryofdifferentialequations,Lipschitzcontinu-ityisthecentralconditionofthePicard–Lindelöfthe-oremwhichguaranteestheexistenceanduniquenessofthesolutiontoaninitialvalueproblem.AspecialtypeofLipschitzcontinuity,calledcontraction,isusedintheBanachfixedpointtheorem.Wehavethefollowingchainofinclusionsforfunctionsoveraclosedandbounded[2]subsetofthereallineContinuouslydifferentiable⊆Lipschitzcontinuous⊆α-Höldercontinuous⊆uniformlycontinuous=continuouswhere0α≤1.WealsohaveLipschitzcontinuous⊆absolutelycontin-uous⊆boundedvariation⊆differentiablealmosteverywhere1DefinitionsGiventwometricspaces(X,dX)and(Y,dY),wheredXdenotesthemetriconthesetXanddYisthemetriconsetY(forexample,YmightbethesetofrealnumbersRwiththemetricdY(y1,y2)=|y1−y2|,andXmightbeasubsetofR),afunctionf:X→YiscalledLipschitzcontinuousifthereexistsarealconstantK≥0suchthat,forallx1andx2inX,dY(f(x1);f(x2))KdX(x1;x2):[3]AnysuchKisreferredtoasaLipschitzconstantforthefunctionf.Thesmallestconstantissometimescalledthe(best)Lipschitzconstant;however,inmostcases,thelatternotionislessrelevant.IfK=1thefunctioniscalledashortmap,andif0≤K1thefunctioniscalledacontraction.ForaLipschitzcontinuousfunction,thereisadoublecone(showninwhite)whosevertexcanbetranslatedalongthegraph,sothatthegraphalwaysremainsentirelyoutsidethecone.Theinequalityis(trivially)satisfiedifx1=x2.Otherwise,onecanequivalentlydefineafunctiontobeLipschitzcon-tinuousifandonlyifthereexistsaconstantK≥0suchthat,forallx1≠x2,dY(f(x1);f(x2))dX(x1;x2)K:Forreal-valuedfunctionsofseveralrealvariables,thisholdsifandonlyiftheabsolutevalueoftheslopesofallsecantlinesareboundedbyK.ThesetoflinesofslopeKpassingthroughapointonthegraphofthefunctionformsacircularcone,andafunctionisLipschitzifandonlyifthegraphofthefunctioneverywhereliescompletelyout-sideofthiscone(seefigure).AfunctioniscalledlocallyLipschitzcontinuousifforeveryxinXthereexistsaneighborhoodUofxsuchthatfrestrictedtoUisLipschitzcontinuous.Equivalently,ifXisalocallycompactmetricspace,thenfislocallyLip-schitzifandonlyifitisLipschitzcontinuousoneverycompactsubsetofX.Inspacesthatarenotlocallycom-pact,thisisanecessarybutnotasufficientcondition.Moregenerally,afunctionfdefinedonXissaidtobeHöldercontinuousortosatisfyaHölderconditionoforderα0onXifthereexistsaconstantM0suchthatdY(f(x);f(y))MdX(x;y)forallxandyinX.SometimesaHölderconditionoforderαisalsocalledauniformLipschitzconditionof123PROPERTIESorderα0.IfthereexistsaK≥1with1KdX(x1;x2)dY(f(x1);f(x2))KdX(x1;x2)thenfiscalledbilipschitz(alsowrittenbi-Lipschitz).Abilipschitzmappingisinjective,andisinfactahomeomorphismontoitsimage.AbilipschitzfunctionisthesamethingasaninjectiveLipschitzfunctionwhoseinversefunctionisalsoLipschitz.2ExamplesLipschitzcontinuousfunctionsThefunctionf(x)=√x2+5definedforallrealnumbersisLipschitzcontinuouswiththeLipschitzconstantK=1,becauseitiseverywheredifferentiableandtheabsolutevalueofthederivativeisboundedaboveby1.SeethefirstpropertylistedbelowunderProperties.Likewise,thesinefunctionisLipschitzcon-tinuousbecauseitsderivative,thecosinefunc-tion,isboundedaboveby1inabsolutevalue.Thefunctionf(x)=|x|definedontherealsisLipschitzcontinuouswiththeLipschitzcon-stantequalto1,bythereversetriangleinequal-ity.ThisisanexampleofaLipschitzcontin-uousfunctionthatisnotdifferentiable.Moregenerally,anormonavectorspaceisLips-chitzcontinuouswithrespecttotheassociatedmetric,withtheLipschitzconstantequalto1.Lipschitzcontinuousfunctionsthatarenotevery-wheredifferentiableThefunctionf(x)=|x|.Continuousfunctionsthatarenot(globally)Lips-chitzcontinuousThefunctionf(x)=√xdefinedon[0,1]isnotLipschitzcontinuous.Thisfunctionbecomesinfinitelysteepasxapproaches0sinceitsderivativebecomesinfinite.However,itisuniformlycontinuous[4]aswellasHöldercontinuousofclassC0,αforα≤1/2.Differentiablefunctionsthatarenot(globally)Lips-chitzcontinuousThefunctionf(x)=x3/2sin(1/x)wherex≠0andf(0)=0,restrictedon[0,1],givesanexampleofafunctionthatisdifferentiableonacompactsetwhilenotlocallyLipschitzbecauseitsderivativefunctionisnotbounded.Seealsothefirstpropertybelow.Analyticfunctionsthatarenot(globally)LipschitzcontinuousTheexponentialfunctionbecomesarbitrarilysteepasx→∞,andthereforeisnotglob-allyLipschitzcontinuous,despitebeingananalyticfunction.Thefunctionf(x)=x2withdomainallrealnumbersisnotLipschitzcontinuous.Thisfunctionbecomesarbitrarilysteepasxap-proachesinfinity.ItishoweverlocallyLips-chitzcontinuous.3PropertiesAneverywheredifferentiablefunctiong:R→RisLipschitzcontinuous(withK=sup|g′(x)|)ifandonlyifithasboundedfirstderivative;onedirectionfollowsfr