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

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 :$$displaystyle f: X o Y is called Lipschitz continuous if there exists a real constant "K" ≥ 0 such that, for all "x"₁ and "x"₂ in "X",:$$d_Y(f(x_1), f(x_2)) le K d_X(x_1, x_2). 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 "x"₁ = "x"₂. 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 "x"₁ ≠ "x"₂, :$$frac{d_Y(f(x_1),f(x_2))}{d_X(x_1,x_2)}le K.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.

More generally, a function "f" defined on "X" is said be Hölder continuous or to satisfy a Hölder condition of order "α" > 0 on "X" if there exists a constant "M" > 0 such that:$$displaystyle d_Y(f(x), f(y)) < M d_X(x, y)^{alpha} 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:$$frac{1}{K}d_X(x_1,x_2) le d_Y(f(x_1), f(x_2)) le K d_X(x_1, x_2)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.

Examples

* The function $$f(x)=x^2 with domain all real numbers is "not" Lipschitz continuous. This function becomes arbitrarily steep as $$x o infty. It is however locally Lipschitz continuous.

* The function $$f(x)=x^2 defined on $$ [-3, 7] is Lipschitz continuous, with Lipschitz constant $$K=14.

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

* The function "f"("x")="x^3/2"sin(1/"x") ("x" &ne;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, since its derivative function is not bounded. See also the first property below.

Properties

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

* The Lipschitz property is preserved better than differentiability: if a sequence of Lipschitz continuous functions $${f_k} all having a fixed Lipschitz constant "K" converges to $$f in the infinity norm sense (uniform convergence), then $$f is also Lipschitz continuous with the same Lipschitz constant "K". This essentially means that the metric space of all Lipschitz functions with the infinity norm, is closed.

* The above property is not true for all metrics (for example L1 norm). It is also not true for sequences of Lipschitz continuous functions $${f_k} where each function of the sequence may have an arbitrary Lipschitz constant $$L_k. It is possible to find a sequence of Lipschitz continuous functions that converges to a non Lipschitz continuous function.

* If the sequence $${L_k} is bounded, i.e. $$L_k for all $$k, then it is true that $$f is Lipschitz continuous with a Lipschitz constant equal to (or smaller than) $$L.

*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 $$f:M o N, then the restriction of $$f to any compact set $$A subseteq M is Lipschitz continuous.

*If "U" is a subset of the metric space "M" and "f" : "U" &rarr; R is a Lipschitz continuous map, there always exist Lipschitz continuous maps "M" &rarr; 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" &rarr; 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")| &le; "K" whenever the derivative exists. Conversely, if "f" : "I" &rarr; R is absolutely continuous and thus differentiable almost everywhere, and satisfies |"f’"("x")| &le; "L" for almost all "x" in "I" then "f" is Lipschitz continuous with Lipschitz constant "K" &le; "L". In particular, if "f" is a differentiable map with bounded derivative, |"f’"("x")| &le; "L" for all "x" in "I", then "f" is Lipschitz continuous with Lipschitz constant "K" &le; "L", a consequence of the mean value theorem.

Lipschitz manifolds

Let "U" and "V" be two open sets in Rⁿ. A function "T" : "U" &rarr; "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; ^{[http://eom.springer.de/T/t093230.htm]} it can in that sense 'nearly' be smoothed.

ee also

*Modulus of continuity

References