NationMaster - Encyclopedia: Lipschitz continuity

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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... 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 â€“ October 7, 1903) was a German mathematician and Professor at the University of Bonn from 1864. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... 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. Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... In mathematics, the Picardâ€“LindelÃ¶f theorem or Picards existence theorem on existence and uniqueness of solutions of differential equations (Picard 1890, LindelÃ¶f 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. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In mathematics, a real-valued function f on Rn satisfies a HÃ¶lder condition, or is HÃ¶lder continuous, when there are nonnegative real constants C, Î±, such that, , This condition generalizes to functions between any two metric spaces. ...

1 Definitions
- 1.1 Real numbers
- 1.2 Metric spaces
2 Examples
3 Properties
4 Lipschitz manifold structure
5 See also

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 as numbers that can be given by an infinite decimal representation, such as 2. ...

$f colon D subseteq mathbb{R} to mathbb{R}$

is called Lipschitz continuous or is said to satisfy a Lipschitz condition if there exists a constant $K ge 0$ such that for all $x 1, x 2$ in $D$

$|f(x_1)-f(x_2)|le K |x_1-x_2|.$

The smallest such K is called the Lipschitz constant of the function $f .$

As this equation is immediate if $x 1 = x 2$ , one can equivalently define a function to be Lipschitz if and only if

$frac{|f(x_1)-f(x_2)|}{|x_1-x_2|}le K$

for $x_1neq x_2$ , i.e., iff the slopes of secants are bounded. IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

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. ...

A function $f$ , defined on $[a, b]$ , is said to satisfy a uniform Lipschitz condition of order $α > 0$ on $[a, b]$ if there there exists a constant $M > 0$ such that

| f (x) - f (y) | < M | x - y | α

for all $x$ and $y$ in $[a, b]$ .

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, $U$ is a subset of $M$ , 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 function is a function which defines a distance between elements of a set. ...

$f : U to N$

is called Lipschitz continuous if there exists a constant $K ge 0$ such that for all $x 1$ and $x 2$ in $U$

$d'(f(x_1), f(x_2)) le K d(x_1, x_2).$

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 $K ge 1$ with

$frac{1}{K}d(x_1,x_2) le d'(f(x_1), f(x_2)) le K d(x_1, x_2)$

then $f$ is called bilipschitz (also written bi-Lipschitz): this is an isomorphism in the category of Lipschitz maps.

Examples

The function $f (x) = x 2$ defined on $[ - 3,7]$ is Lipschitz continuous, with Lipschitz constant $K = 14$ . This follows from the last property below.

The function $f(x)=sqrt{x^2+5}$ defined for all real numbers is Lipschitz continuous with the Lipschitz constant $K = 1$ .

The function $f (x) = | x |$ defined on the reals is Lipschitz continuous with the Lipschitz constant equal to 1. This is an example of a Lipschitz continuous function that is not differentiable.

The function $f (x) = x 2$ (the same function as in the first example) with domain all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as $x to infty$ . It is however locally Lipschitz.

The function $f(x)=sqrt{x}$ defined on $[0,1]$ is not Lipschitz continuous. This function becomes infinitely steep as $x to 0$ since its derivative becomes infinite. It is however Hölder continuous of class $C 0,α$ , for $alpha leq 1/2$ .

In mathematics, a real-valued function f on Rn satisfies a HÃ¶lder condition, or is HÃ¶lder continuous, when there are nonnegative real constants C, Î±, such that, , This condition generalizes to functions between any two metric spaces. ...

Properties

An everywhere differentiable function g is Lipschitz continuous (with $C = sup | g'(x) |$ ) if it has bounded first derivative; one direction follows from the mean value theorem. Thus any $C 1$ function is locally Lipschitz, as continuous functions on a locally compact space are locally bounded.

The Lipschitz properties is preserved better than differentiability: if a sequence of Lipschitz continuous functions ${f k}$ converges to $f$ , then $f$ is also Lipschitz continuous.

Every Lipschitz continuous map is uniformly continuous, and hence a fortiori continuous.

Every bilipschitz function (see definition above) is injective. A bilipschitz function is the same thing as a Lipschitz bijection whose inverse function is also Lipschitz.

Given a locally Lipschitz continuous function $f:M to N$ , then the restriction of $f$ to any compact set $A subseteq M$ is Lipschitz continuous.

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).

Rademacher's theorem states that a Lipschitz continuous map f : I → R, where I is an interval in R, is almost everywhere differentiable (that is, it is 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, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ... 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. ... 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. ... 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. ... 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... In mathematical analysis, Rademachers theorem states the following: If A is an open subset of Rn and f : A â†’ Rm is Lipschitz, then f is differentiable almost everywhere in A. Juha Heinonen, Lectures on Lipschitz Analysis, Lectures at the 14th JyvÃ¤skylÃ¤ Summer School in August 2004. ... 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. ... For a non-technical overview of the subject, see Calculus. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ...

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 piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. ... 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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