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In mathematics, more specifically in real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions which is stronger than regular continuity. Intuitively, a Lipschitz continuous function is limited in how fast it can change; a line joining any two points on the graph of this function will never have a slope steeper than a certain number called the Lipschitz constant of the function. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...
Rudolf Otto Sigismund Lipschitz (May 14, 1832 – October 7, 1903) was a German mathematician and Professor at the University of Bonn from 1864. ...
Partial plot of a function f. ...
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
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. An illustration of a differential equation. ...
In mathematics, the Picard–Lindelöf theorem or Picards existence theorem on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem has exactly one solution if f is Lipschitz continuous in , continuous in as long as stays bounded. ...
In mathematics, an initial value problem is a statement of a differential equation together with specified value of the unknown function at a given point in the domain of the solution. ...
In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some real number such that, for all x and y in M, The smallest such value of k is called the Lipschitz constant...
The Banach fixed point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those...
The concept of Lipschitz continuity can be defined on metric spaces and thus also on normed vector spaces. A generalisation of Lipschitz continuity is called Hölder continuity. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
Link titleIn mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...
Definitions
Real numbers A real valued function f defined on a subset D of the real numbers In mathematics, the real numbers may be described informally in several different ways. ...
is called Lipschitz continuous or is said to satisfy a Lipschitz condition if there exists a constant such that for all x,y in D The smallest such K is called the Lipschitz constant of the function f.
The function is called locally Lipschitz continuous if for every x in D there exists a neighborhood U(x) so that f restricted to U is Lipschitz continuous. In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...
Metric spaces Given two metric spaces (M,d) and (N,d'), where d and d' denotes the metric on the sets M and N respectively, a function In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
In mathematics a metric or distance is a function which assigns a distance to elements of a set. ...
is called Lipschitz continuous if there exists a constant such that for all x and y in M The smallest such K is called the Lipschitz constant of the function f. If K = 1 the function is called short map, if K < 1 the function is called contraction. In mathematics, a short map is a function f from a metric space X to another metric space Y such that for any we have . Here and denote metrics on and , respectively. ...
In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some real number such that, for all x and y in M, The smallest such value of k is called the Lipschitz constant...
If there exists a with
then f is called bilipschitz.
Examples - The function f(x) = x2 defined on [ − 3,7] is Lipschitz continuous, with Lipschitz constant K=14. This follows from the observation above.
- The function defined for all real numbers is Lipschitz continuous with the Lipschitz constant K=1.
- The function f(x) = 2 | x − 3 | defined on [ − 10,10] is Lipschitz continuous with the Lipschitz constant equal to 2. This is an example of a Lipschitz continuous function that is not differentiable.
- The function f(x) = x2 (the same function as in the first example) defined for all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as x→∞.
- The function defined on [0,3] is not Lipschitz continuous. This function becomes infinitely steep as x→0 since its derivative becomes infinite.
In mathematics, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ...
For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c. ...
Properties Every Lipschitz continuous map is uniformly continuous, and hence a fortiori continuous. In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but...
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
Every bilipschitz function is injective. A bilipschitz function is the same thing as a Lipschitz bijection whose inverse function is also Lipschitz. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
A bijective function. ...
In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...
Given a locally Lipschitz continuous function , then the restriction of f to any compact set is Lipschitz continuous. In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ...
If U is a subset of the metric space M and f : U → R is a Lipschitz continuous map, there always exist Lipschitz continuous maps M → R which extend f and have the same Lipschitz constant as f (see also Kirszbraun theorem). In mathematics, the Kirszbraun theorem in mathematical analysis states that if U is a subset of Euclidean space En and f : U → Em is a Lipschitz-continuous map, then there is a Lipschitz-continuous map F: En → Em that extends f, and has the same Lipschitz constant as...
Rademacher's theorem states that a Lipschitz continuous map f : I → R, where I is an interval in R, is almost everywhere differentiable (everywhere except on a set of Lebesgue measure 0). If K is the Lipschitz constant of f, then |f’(x)| ≤ K whenever the derivative exists. Conversely, if f : I → R is a differentiable map with bounded derivative, |f’(x)| ≤ L for all x in I, then f is Lipschitz continuous with Lipschitz constant K ≤ L, a consequence of the mean value theorem. In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...
In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...
In mathematics, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ...
In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ...
For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c. ...
Lipschitz manifold structure There is a notion of a Lipschitz structure on a topological manifold, since there is a pseudogroup structure on Lipschitz homeomorphisms. This structure is intermediate between that of a piecewise-linear manifold and a smooth manifold. In fact a PL structure gives rise to a unique Lipschitz structure[1]; it can in that sense 'nearly' be smoothed. In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...
In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra (such as quasigroup, for example). ...
In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...
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