Lipschitz continuity
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 uniform continuity).For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous.[1] In the theory of differential equations, 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, is used in the Banach fixed-point theorem.[2] We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line: whereWe also have Given two metric spaces (X, dX) and (Y, dY), where dX denotes the metric on the set X and dY 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 x1 and x2 in X, Any such K is referred to as a Lipschitz constant for the function f and f may also be referred to as K-Lipschitz.[9] Let F(x) be an upper semi-continuous function of x, and that F(x) is a closed, convex set for all x.
mathematical analysisGermanmathematicianRudolf Lipschitzuniform continuityfunctionscontinuous functionabsolute valuemodulus of uniform continuitydifferential equationsPicard–Lindelöf theoreminitial value problemcontractionBanach fixed-point theoremclosed and boundedContinuously differentiableHölder continuousabsolutely continuousuniformly continuousmetric spacesmetricshort mapreal-valued functionreal numbersif and only ifneighborhoodlocally compactHölder conditioninjectivehomeomorphisminverse functiondifferentiablereverse triangle inequalityexponential functionanalytic functionfirst derivativemean value theoremalmost everywhereLebesgue measureessentially boundedRademacher's theoremtotal derivativeuniformlyBanach spaceStone–Weierstrass theoremWeierstrass approximation theoremcontinuousequicontinuousArzelà–Ascoli theoremuniformly boundedKirszbraun theoremtopological manifoldatlas of chartspseudogroupsmooth manifoldspiecewise-linear manifoldupper semi-continuousContraction mappingDini continuityModulus of continuityQuasi-isometryJohnson–Lindenstrauss lemmaSpringer-VerlagGromov, MikhaelRosenberg, JonathanAustralian National UniversityEncyclopedia of MathematicsEMS PressMetric spaceCauchy sequenceCompletenessEquivalent metricsMetrizable spaceTriangle inequalityBaire category theoremBanach fixed-pointKuratowski embeddingLebesgue's number lemmaMetrization theoremsNagata–SmirnovUrysohn'sMetric mapDilationEquicontinuityQuasi-IsometryMetric derivativeMetric outer measureMetric projectionMotionQuasisymmetricStretch factorIsomorphismUniform convergenceCompleteConvexDoublingHyperbolicLength metric spaceMetric space aimed at its subspacePolishTotally boundedTree-gradedUltrametric spaceUniformly disconnectedUrysohn universalBoundedDeloneDiameterDistance setGromov productGromov–Hausdorff convergenceHausdorff distanceKuratowski convergencePacking dimensionPorousPositively separated setsTight spanManifoldsEuclidean distanceRiemannianFunctional analysisMeasure theoryChebyshev distanceInner product spaceLévy metric