Lipschitz continuity

For a Lipschitz continuous function there is a double cone (white) whose origin can be moved along the graph so that it always remains outside the double cone

The Lipschitz , even stretching limitations , is a term from the mathematical branch of Analysis . It is a property of a function , therefore one speaks mostly of lipschitzstetigen functions (or of Lipschitz continuous functions ). The Lipschitz continuity is a tightening of the continuity . This property is named after the mathematician Rudolf Lipschitz .

To put it clearly, a Lipschitz continuous function can change only to a limited extent quickly: All secants of a function have a slope , the amount of which is not greater than the Lipschitz constant . The set of all Lipschitz continuous functions is called the Lipschitz space . Generalizations of the Lipschitz continuity are the Hölder continuity and the local Hölder continuity .

definition

A function is called lipschitz continuous if there is a constant such that ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} }$ ${\displaystyle L}$

${\displaystyle |f(x_{1})-f(x_{2})|\leq L\cdot |x_{1}-x_{2}|}$

applies to all . ${\displaystyle x_{1},x_{2}\in \mathbb {R} }$

This is a special case of the following general definition.

Be and metric spaces . A function is called lipschitz continuous if there is a real number such that ${\displaystyle (X,d_{X})}$ ${\displaystyle (Y,d_{Y})}$ ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle L}$

${\displaystyle \forall x_{1},x_{2}\in X:d_{Y}(f(x_{1}),f(x_{2}))\leq L\cdot d_{X}(x_{1},x_{2})}$

is satisfied. is called the Lipschitz constant and it always applies . In clear terms, the amount of the slope is limited from above by . If a function is lipschitz continuous, one also says that it fulfills the Lipschitz condition . ${\displaystyle L}$ ${\displaystyle L\geq 0}$ ${\displaystyle f}$ ${\displaystyle L}$

A weakening of the Lipschitz continuity is the local Lipschitz continuity. A function is locally called lipschitz continuous if there is an environment around every point , so that the restriction of lipschitz to this environment is lipschitz continuous. A function that is only defined on a subset is called lipschitz- or locally lipschitz-continuous if it is lipschitz- or locally lipschitz-continuous with respect to the metric spaces and . ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle A\subset X}$ ${\displaystyle (A,d_{X}|_{A})}$ ${\displaystyle (Y,d_{Y})}$

properties

Lipschitz continuous functions are locally lipschitz continuous (choose completely as environment and always as Lipschitz constant ). Locally Lipschitz continuous functions are continuous (choose in the - definition of continuity), and accordingly Lipschitz continuous functions are uniformly continuous . Hence, Lipschitz continuity is "stronger" than uniform continuity. The reverse is generally not true. B. the function is Hölder continuous with exponents and therefore uniformly continuous, but not Lipschitz continuous (see example). ${\displaystyle X}$ ${\displaystyle L}$ ${\displaystyle \delta =\varepsilon /L}$ ${\displaystyle \varepsilon }$ ${\displaystyle \delta }$ ${\displaystyle f\colon [0,1]\rightarrow \mathbb {R} ,~x\mapsto {\sqrt {x}}}$ ${\displaystyle 1/2}$

According to Rademacher's theorem , a Lipschitz continuous function can be differentiated almost everywhere . However, there are also functions that are differentiable but not lipschitz continuous, e.g. B. . A differentiable function with is lipschitz continuous if and only if its first derivative is bounded. ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} ,~x\mapsto x^{2}}$ ${\displaystyle f\colon (a,b)\rightarrow \mathbb {R} }$ ${\displaystyle a,b\in \mathbb {R} \cup \{\pm \infty \}}$

Examples

• For a lipschitz continuous function is the quotient ${\displaystyle f\colon (X,d_{X})\rightarrow (Y,d_{Y})}$

${\displaystyle {\frac {d_{Y}(f(x_{1}),f(x_{2}))}{d_{X}(x_{1},x_{2})}}}$

with bounded from above by every Lipschitz constant . For locally Lipschitz continuous functions the quotient is restricted to sufficiently small surroundings. ${\displaystyle x_{1}\neq x_{2}\in X}$ ${\displaystyle f}$

Therefore, the function is with due ${\displaystyle f\colon [0,1]\to \mathbb {R} }$ ${\displaystyle x\mapsto {\sqrt {x}}}$

${\displaystyle {\frac {|f(x_{1})-f(0)|}{|x_{1}-0|}}={\frac {1}{{\sqrt {x}}_{1}}}\,{\xrightarrow {x_{1}\searrow 0}}\,\infty }$

admittedly steady and even uniformly steady, but not locally lipschitz continuous and consequently not lipschitz continuous either.

• For the function with follows with ${\displaystyle g\colon [a,b]\to \mathbb {R} }$ ${\displaystyle g(x)=x^{2}}$

${\displaystyle L:=\max _{x_{1},x_{2}\in [a,b]}(|x_{1}+x_{2}|)=2\max {(|a|,|b|)}}$,

that . ${\displaystyle |g(x_{1})-g(x_{2})|=|x_{1}^{2}-x_{2}^{2}|=|x_{1}+x_{2}|\cdot |x_{1}-x_{2}|\leq L\cdot |x_{1}-x_{2}|}$

That is, a Lipschitz constant for this function is on the interval . ${\displaystyle L}$ ${\displaystyle \left[a,b\right]}$

Because the quotient is the same , it follows that Lipschitz is only continuous for a restricted domain, but not for an unrestricted domain. The function also defined by is therefore not lipschitz continuous. ${\displaystyle g}$ ${\displaystyle |x_{1}+x_{2}|}$ ${\displaystyle g}$ ${\displaystyle g(x)=x^{2}}$ ${\displaystyle g\colon \mathbb {R} \to \mathbb {R} }$

• The amount function , defined as ${\displaystyle h\colon \mathbb {R} \rightarrow \mathbb {R} }$

${\displaystyle h(x)=|x|}$,

is lipschitz continuous with because of the inverted triangle inequality , but it is not differentiable (at that point ). ${\displaystyle {\bigl |}|x_{1}|-|x_{2}|{\bigr |}\leq |x_{1}-x_{2}|}$ ${\displaystyle L=1}$ ${\displaystyle x=0}$

application

Lipschitz continuity is an important concept in the theory of ordinary differential equations to prove existence and uniqueness of solutions (see Picard-Lindelöf theorem ). Self-images with a Lipschitz constant less than one are called contractions . These are important for Banach's Fixed Point Theorem .

In the theory of partial differential equations are Lipschitz areas considered. These have the property that their edge, called the Lipschitz edge , can be described locally by a Lipschitz continuous function.

Lipschitz room

If (or more generally a metric space), the set of real-valued Lipschitz continuous functions is sometimes denoted by. ${\displaystyle X\subseteq \mathbb {R} }$ ${\displaystyle \left(X,\,d_{X}\right)}$ ${\displaystyle X}$ ${\displaystyle \mathrm {Lip} \left(X\right)}$

For (or more generally for with the Euclidean metric ) every affine-linear function is Lipschitz continuous. After all, in a general metric space all constant functions are lipschitz continuous. In particular, it is not empty and contains the constant null function . ${\displaystyle X\subseteq \mathbb {R} }$ ${\displaystyle X\subseteq \mathbb {R} ^{n}}$ ${\displaystyle \mathrm {Lip} \left(X\right)}$

Are and so applies as well . This is a real vector space , a function space . ${\displaystyle f,\,g\in \mathrm {Lip} \left(X\right)}$ ${\displaystyle \lambda \in \mathbb {R} }$ ${\displaystyle \lambda \,f\in \mathrm {Lip} \left(X\right)}$ ${\displaystyle f+g\in \mathrm {Lip} \left(X\right)}$ ${\displaystyle \mathrm {Lip} \left(X\right)}$

If the quantity is also limited , the same applies to the point-by-point product . This becomes a function algebra . ${\displaystyle X}$ ${\displaystyle f\cdot g\in \mathrm {Lip} \left(X\right)}$ ${\displaystyle \mathrm {Lip} \left(X\right)}$

See also

• Kirszbraun's theorem on the continuability of Lipschitz continuous functions

literature

• Harro Heuser: Textbook of Analysis - Part 1 , 6th edition, Teubner 1989, ISBN 3-519-42221-2 , pp. 136, 212
• Konrad Königsberger: Analysis 1 . 2nd edition, Springer 1992, ISBN 3-540-55116-6 , p. 80
• Wolfgang Walter: Analysis 1 . 7th edition, Springer 2004, ISBN 978-3-540-35078-1 , pp. 44, 45

Web links

Wikibooks: Math for non-freaks: Lipschitz continuity  - learning and teaching materials

• Lipschitz condition in the Encyclopaedia of Mathematics (accessed December 2, 2009)
• Lipschitz continuous on PlanetMath (accessed December 2, 2009)

Individual evidence

1. ^ Walter Rudin : Functional Analysis. McGraw-Hill, New York 1991. ISBN 0070542368 , pp. 41, 420.