In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. 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 (or modulus of uniform continuity ). For instance, every function that has bounded first derivatives is Lipschitz continuous.
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.
We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line
where 0 < α ≤ 1. We 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. The smallest constant is sometimes called the (best) Lipschitz constant; however, in most cases, the latter notion is less relevant. If K = 1 the function is called a short map , and if 0 ≤ K < 1 and f maps a metric space to itself, the function is called a contraction .
In particular, a real-valued function f : R → R is called Lipschitz continuous if there exists a positive real constant K such that, for all real x1 and x2,
In this case, Y is the set of real numbers R with the standard metric dY(y1, y2) = |y1 − y2|, and X is a subset of R.
In general, the inequality is (trivially) satisfied if x1 = x2. Otherwise, one can equivalently define a function to be Lipschitz continuous if and only if there exists a constant K ≥ 0 such that, for all x1 ≠ x2,
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 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 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.
If there exists a K≥ 1 with
then f is called bilipschitz (also written bi-Lipschitz). A bilipschitz mapping is injective, and is in fact a homeomorphism onto its image. A bilipschitz function is the same thing as an injective Lipschitz function whose inverse function is also Lipschitz.
Let U and V be two open sets in Rn. A function T : U → V is called bi-Lipschitz if it is a Lipschitz homeomorphism onto its image, and its inverse is also Lipschitz.
Using bi-Lipschitz mappings, it is possible to define a Lipschitz structure on a topological manifold, since there is a pseudogroup structure on bi-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;it can in that sense 'nearly' be smoothed.
Let F(x) be an upper semi-continuous function of x, and that F(x) is a closed, convex set for all x. Then F is one-sided Lipschitzif
for some C and for all x1 and x2.
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.
In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some nonnegative real number such that for all x and y in M,
In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, where the maximum distance between f(x) and f(y) may depend on x and y themselves.
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
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 operations of calculus—differentiation and integration. This relationship is commonly characterized in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure.
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.
In functional analysis, a bounded linear operator is a linear transformation L : X → Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces, then L is bounded if and only if there exists some M ≥ 0 such that for all x in X,
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 families, and thus sequences of functions.
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 interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.
In mathematical analysis, a modulus of continuity is a function ω : [0, ∞] → [0, ∞] 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
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered "smooth" if it is differentiable everywhere. At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function.
In geometric topology, Busemann functions are used to study the large-scale geometry of geodesics in Hadamard spaces and in particular Hadamard manifolds. They are named after Herbert Busemann, who introduced them; he gave an extensive treatment of the topic in his 1955 book "The geometry of geodesics".
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
In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev.
In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for Henri Lebesgue.
In mathematics, Gårding's inequality is a result that gives a lower bound for the bilinear form induced by a real linear elliptic partial differential operator. The inequality is named after Lars Gårding.