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.

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.

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.

Definitions

For a Lipschitz continuous function, there is a double cone (shown in green) whose vertex can be translated along the graph, so that the graph always stays entirely inside the cone.

Given two metric spaces (X, d_X) and (Y, d_Y), where d_X denotes the metric on the set X and d_Y is the metric on set Y (for example, Y might be the set of real numbers R with the metric d_Y(x, y) = |x − y|, and X might be a subset of R), a function

is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all x₁ and x₂ in X,

The smallest such K is called the Lipschitz constant of the function ƒ. If K = 1 the function is called a short map, and if 0 < K < 1 the function is called a contraction.

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 a real argument, this holds iff the slopes of all secant lines are bounded.

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

Examples

Lipschitz continuous functions

• The function f(x) = √x² + 5 defined for all real numbers is Lipschitz continuous with the Lipschitz constant K = 1, because it is everywhere differentiable and the absolute value of the derivative is bounded above by 1.

• Likewise, the sine function is Lipschitz continuous because its derivative, the cosine function, is bounded above by 1 in absolute value.

• The function f(x) = |x| defined on the reals is Lipschitz continuous with the Lipschitz constant equal to 1, by the reverse triangle inequality. This is an example of a Lipschitz continuous function that is not differentiable. More generally, a norm on a vector space is Lipschitz continuous with respect to the associated metric, with the Lipschitz constant equal to 1.

Continuous functions that are not (globally) Lipschitz continuous

• The function f(x) = x² with domain all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as x approaches infinity. It is however locally Lipschitz continuous.

• The function f(x) = √x defined on [0, 1] is not Lipschitz continuous. This function becomes infinitely steep as x approaches 0 since its derivative becomes infinite. It is however Hölder continuous of class C^0, α, for α ≤ 1/2.

Differentiable functions that are not (globally) Lipschitz continuous

• The function f(x) = x^3/2sin(1/x) (x ≠ 0) and f(0) = 0, restricted on [0, 1], gives an example of a function that is differentiable on a compact set while not locally Lipschitz because its derivative function is not bounded. See also the first property below.

Properties

Lipschitz continuity of functions on the real line is closely related to differentiability. An everywhere differentiable function g : R → R is Lipschitz continuous (with K = sup |g′(x)|) if and only if it has bounded first derivative; one direction follows from the mean value theorem. In particular, any C¹ function is locally Lipschitz, as continuous functions on a locally compact space are locally bounded.

A Lipschitz function g : R → R is absolutely continuous and therefore is differentiable almost everywhere, that is, differentiable at every point outside a set of Lebesgue measure zero. Its derivative is essentially bounded 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 ƒ : I → R is absolutely continuous and thus differentiable almost everywhere, and satisfies |ƒ′(x)| ≤ K for almost all x in I, then ƒ is Lipschitz continuous with Lipschitz constant at most K.

More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ƒ : U → R^m, where U is an open set in Rⁿ, is almost everywhere differentiable. Moreover, if K is the Lipschitz constant of ƒ, then whenever the total derivative Dƒ exists.

The Lipschitz property is preserved better than differentiability: suppose that {f_n} is a sequence of Lipschitz continuous mappings between two metric spaces, and that all f_n have Lipschitz constant bounded by some K. If ƒ_n converges to a mapping ƒ uniformly, then ƒ 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 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 dense in the Banach space of continuous functions, an elementary consequence of the Stone–Weierstrass theorem.

Every Lipschitz continuous map is uniformly continuous, and hence a fortiori continuous. More generally, a set of functions with bounded Lipschitz constant forms an equicontinuous set. The Arzelà–Ascoli theorem implies that if {f_n} is a uniformly bounded 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).

If U is a subset of the metric space M and ƒ : U → R is a Lipschitz continuous map, there always exist Lipschitz continuous maps M → R which extend ƒ and have the same Lipschitz constant as ƒ (see also Kirszbraun theorem).

Lipschitz manifolds

Let U and V be two open sets in Rⁿ. A function T : U → V is called bi-Lipschitz if it is a 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;^[1] it can in that sense 'nearly' be smoothed.

See also

• Hölder continuous
• Modulus of continuity

References