In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, Lipschitz continuity, named after
German
German(s) may refer to:
* Germany (of or related to)
** Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Rudolf Lipschitz
Rudolf Otto Sigismund Lipschitz (14 May 1832 – 7 October 1903) was a German mathematician who made contributions to mathematical analysis (where he gave his name to the Lipschitz continuity condition) and differential geometry, as well as numbe ...
, is a strong form of
uniform continuity
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
for
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s. Intuitively, a Lipschitz
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
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
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
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 (or ''
modulus of uniform continuity''). For instance, every function that has bounded first derivatives is Lipschitz continuous.
In the theory of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s, Lipschitz continuity is the central condition of the
Picard–Lindelöf theorem
In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauc ...
which guarantees the existence and uniqueness of the solution to an
initial value problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or oth ...
. A special type of Lipschitz continuity, called
contraction
Contraction may refer to:
Linguistics
* Contraction (grammar), a shortened word
* Poetic contraction, omission of letters for poetic reasons
* Elision, omission of sounds
** Syncope (phonology), omission of sounds in a word
* Synalepha, merged ...
, is used in the
Banach fixed-point theorem
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certa ...
.
We have the following chain of strict inclusions for functions over a
closed and bounded non-trivial interval of the real line:
:
Continuously differentiable ⊂ Lipschitz continuous ⊂
-
Hölder continuous Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of number ...
,
where
. We also have
: Lipschitz continuous ⊂
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
⊂
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
.
Definitions
Given two
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s (''X'', ''d''
''X'') and (''Y'', ''d''
''Y''), where ''d''
''X'' denotes the
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
on the set ''X'' and ''d''
''Y'' 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 ''x''
1 and ''x''
2 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. The smallest constant is sometimes called the (best) Lipschitz constant of ''f'' or the dilation or dilatation of ''f''. If ''K'' = 1 the function is called a
short map In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous).
These maps are the morphisms in the category of metric spaces, Met (Isbell 1 ...
, and if 0 ≤ ''K'' < 1 and ''f'' maps a metric space to itself, the function is called a
contraction
Contraction may refer to:
Linguistics
* Contraction (grammar), a shortened word
* Poetic contraction, omission of letters for poetic reasons
* Elision, omission of sounds
** Syncope (phonology), omission of sounds in a word
* Synalepha, merged ...
.
In particular, a
real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called ''real f ...
''f'' : ''R'' → ''R'' is called Lipschitz continuous if there exists a positive real constant K such that, for all real ''x''
1 and ''x''
2,
:
In this case, ''Y'' is the set of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s R with the standard metric ''d''
''Y''(''y
1'', ''y
2'') = , ''y
1'' − ''y
2'', , and ''X'' is a subset of R.
In general, the inequality is (trivially) satisfied if ''x''
1 = ''x''
2. Otherwise, one can equivalently define a function to be Lipschitz continuous
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
there exists a constant ''K'' ≥ 0 such that, for all ''x''
1 ≠ ''x''
2,
:
For real-valued functions of several real variables, this holds if and only if the absolute value of the slopes of all secant lines are bounded by ''K''. The set of lines of slope ''K'' passing through a point on the graph of the function forms a circular cone, and a function is Lipschitz if and only if the graph of the function everywhere lies completely outside of this cone (see figure).
A function is called locally Lipschitz continuous if for every ''x'' in ''X'' there exists a
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
''U'' of ''x'' such that ''f'' restricted to ''U'' is Lipschitz continuous. Equivalently, if ''X'' is a
locally compact metric space, then ''f'' is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of ''X''. In spaces that are not locally compact, this is a necessary but not a sufficient condition.
More generally, a function ''f'' defined on ''X'' is said to be Hölder continuous or to satisfy a
Hölder condition
In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', α > 0, such that
: , f(x) - f(y) , \leq C\, ...
of order α > 0 on ''X'' if there exists a constant ''M'' ≥ 0 such that
:
for all ''x'' and ''y'' in ''X''. Sometimes a Hölder condition of order α is also called a uniform Lipschitz condition of order α > 0.
For a real number ''K'' ≥ 1, if
:
then ''f'' is called ''K''-bilipschitz (also written ''K''-bi-Lipschitz). We say ''f'' is bilipschitz or bi-Lipschitz to mean there exists such a ''K''. A bilipschitz mapping is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
, and is in fact a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
onto its image. A bilipschitz function is the same thing as an injective Lipschitz function whose
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X\t ...
is also Lipschitz.
Examples
;Lipschitz continuous functions:
;Lipschitz continuous functions that are not everywhere differentiable:
;Lipschitz continuous functions that are everywhere differentiable but not continuously differentiable:
;Continuous functions that are not (globally) Lipschitz continuous:
;Differentiable functions that are not (locally) Lipschitz continuous:
;Analytic functions that are not (globally) Lipschitz continuous:
Properties
*An everywhere differentiable function ''g'' : R → R is Lipschitz continuous (with ''K'' = sup , ''g''′(''x''), ) if and only if it has bounded
first derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
; one direction follows from the
mean value theorem. In particular, any continuously differentiable function is locally Lipschitz, as continuous functions are locally bounded so its gradient is locally bounded as well.
*A Lipschitz function ''g'' : R → R is
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
and therefore is differentiable
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
, that is, differentiable at every point outside a set of
Lebesgue measure zero. Its derivative is
essentially bounded
Essence ( la, essentia) is a polysemic term, used in philosophy and theology as a designation for the property or set of properties that make an entity or substance what it fundamentally is, and which it has by necessity, and without which it ...
in magnitude by the Lipschitz constant, and for ''a'' < ''b'', the difference ''g''(''b'') − ''g''(''a'') is equal to the integral of the derivative ''g''′ on the interval
'a'', ''b''
**Conversely, if ''f'' : ''I'' → R is absolutely continuous and thus differentiable almost everywhere, and satisfies , ''f′''(''x''), ≤ ''K'' for almost all ''x'' in ''I'', then ''f'' is Lipschitz continuous with Lipschitz constant at most ''K''.
**More generally,
Rademacher's theorem In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If is an open subset of and is Lipschitz continuous, then is differentiable almost everywhere in ; that is, the points in at which is ''not'' d ...
extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ''f'' : ''U'' → R
''m'', where ''U'' is an open set in R
''n'', is
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
. Moreover, if ''K'' is the best Lipschitz constant of ''f'', then
whenever the
total derivative
In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with resp ...
''Df'' exists.
*For a differentiable Lipschitz map
the inequality
holds for the best Lipschitz constant
of
. If the domain
is convex then in fact
.
*Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all ''f
n'' have Lipschitz constant bounded by some ''K''. If ''f
n'' converges to a mapping ''f''
uniformly, then ''f'' is also Lipschitz, with Lipschitz constant bounded by the same ''K''. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
of continuous functions. This result does not hold for sequences in which the functions may have ''unbounded'' Lipschitz constants, however. In fact, the space of all Lipschitz functions on a compact metric space is a subalgebra of the Banach space of continuous functions, and thus dense in it, an elementary consequence of the
Stone–Weierstrass theorem (or as a consequence of
Weierstrass approximation theorem
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
, because every polynomial is locally Lipschitz continuous).
*Every Lipschitz continuous map is
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
, and hence ''
a fortiori
''Argumentum a fortiori'' (literally "argument from the stronger eason) (, ) is a form of argumentation that draws upon existing confidence in a proposition to argue in favor of a second proposition that is held to be implicit in, and even more cer ...
''
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
. More generally, a set of functions with bounded Lipschitz constant forms an
equicontinuous
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable fa ...
set. The
Arzelà–Ascoli theorem
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interv ...
implies that if is a
uniformly bounded
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family.
...
sequence of functions with bounded Lipschitz constant, then it has a convergent subsequence. By the result of the previous paragraph, the limit function is also Lipschitz, with the same bound for the Lipschitz constant. In particular the set of all real-valued Lipschitz functions on a compact metric space ''X'' having Lipschitz constant ≤ ''K'' is a
locally compact convex subset of the Banach space ''C''(''X'').
*For a family of Lipschitz continuous functions ''f''
α with common constant, the function
(and
) is Lipschitz continuous as well, with the same Lipschitz constant, provided it assumes a finite value at least at a point.
*If ''U'' is a subset of the metric space ''M'' and ''f'' : ''U'' → R is a Lipschitz continuous function, 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, specifically real analysis and functional analysis, the Kirszbraun theorem states that if is a subset of some Hilbert space , and is another Hilbert space, and
: f: U \rightarrow H_2
is a Lipschitz-continuous map, then there ...
). An extension is provided by
::
:where ''k'' is a Lipschitz constant for ''f'' on ''U''.
Lipschitz manifolds
A Lipschitz structure on a
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
is defined using an
atlas of charts whose transition maps are bilipschitz; this is possible because bilipschitz maps form a
pseudogroup In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Lie ...
. Such a structure allows one to define locally Lipschitz maps between such manifolds, similarly to how one defines smooth maps between
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s: if and are Lipschitz manifolds, then a function
is locally Lipschitz if and only if for every pair of coordinate charts
and
, where and are open sets in the corresponding Euclidean spaces, the composition
is locally Lipschitz. This definition does not rely on defining a metric on or .
[ ]
This structure is intermediate between that of a
piecewise-linear manifold
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise line ...
and a
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
: a PL structure gives rise to a unique Lipschitz structure. While Lipschitz manifolds are closely related to topological manifolds,
Rademacher's theorem In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If is an open subset of and is Lipschitz continuous, then is differentiable almost everywhere in ; that is, the points in at which is ''not'' d ...
allows one to do analysis, yielding various applications.
One-sided Lipschitz
Let ''F''(''x'') be an
upper semi-continuous
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, r ...
function of ''x'', and that ''F''(''x'') is a closed, convex set for all ''x''. Then ''F'' is one-sided Lipschitz
if
:
for some ''C'' and for all ''x''
1 and ''x''
2.
It is possible that the function ''F'' could have a very large Lipschitz constant but a moderately sized, or even negative, one-sided Lipschitz constant. For example, the function
:
has Lipschitz constant ''K'' = 50 and a one-sided Lipschitz constant ''C'' = 0. An example which is one-sided Lipschitz but not Lipschitz continuous is ''F''(''x'') = ''e''
−''x'', with ''C'' = 0.
See also
*
*
Dini continuity In mathematical analysis, Dini continuity is a refinement of continuity. Every Dini continuous function is continuous. Every Lipschitz continuous function is Dini continuous.
Definition
Let X be a compact subset of a metric space (such as \mathbb ...
*
Modulus of continuity In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if
:, f(x)-f ...
*
Quasi-isometry In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. ...
*
Johnson-Lindenstrauss lemma – For any integer ''n''≥0, any finite subset ''X''⊆''R
n'', and any real number 0<ε<1, there exists a (1+ε)-bi-Lipschitz function
where
References
{{reflist
Structures on manifolds