Lipschitz continuity

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For a Lipschitz continuous function, there exists a double cone (white) whose origin can be moved along the graph so that the whole graph always stays outside the double cone Lipschitz Visualisierung.gif
For a Lipschitz continuous function, there exists a double cone (white) whose origin can be moved along the graph so that the whole graph always stays outside the double cone

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. [1]

Contents

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. [2]

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 continuousα-Hölder continuous

where 0 < α ≤ 1. We also have

Lipschitz continuous absolutely continuous .

Definitions

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

[3]

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 : RR 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) = |y1y2|, 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 x1x2,

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.

Examples

Lipschitz continuous functions
  • The function 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. See the first property listed below under "Properties".
  • 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.
Lipschitz continuous functions that are not everywhere differentiable
  • The function
Lipschitz continuous functions that are everywhere differentiable but not continuously differentiable
  • The function , whose derivative exists but has an essential discontinuity at .
Continuous functions that are not (globally) 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. However, it is uniformly continuous, [4] and both Hölder continuous of class C0, α for α  1/2 and also absolutely continuous on [0, 1] (both of which imply the former).
Differentiable functions that are not (locally) Lipschitz continuous
  • The function f defined by f(0) = 0 and f(x) = x3/2sin(1/x) for 0<x≤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.
Analytic functions that are not (globally) Lipschitz continuous
  • The exponential function becomes arbitrarily steep as x → ∞, and therefore is not globally Lipschitz continuous, despite being an analytic function.
  • 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.

Properties

where k is a Lipschitz constant for f on U.

Lipschitz manifolds

Let U and V be two open sets in Rn. A function T : UV 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; [5] it can in that sense 'nearly' be smoothed.

One-sided Lipschitz

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 Lipschitz [6] if

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) = ex, with C = 0.

See also

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References

  1. Sohrab, H. H. (2003). Basic Real Analysis. Vol. 231. Birkhäuser. p. 142. ISBN   0-8176-4211-0.
  2. Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2001). Elementary Real Analysis. Prentice-Hall. p. 623.
  3. Searcóid, Mícheál Ó (2006), "Lipschitz Functions", Metric Spaces, Springer undergraduate mathematics series, Berlin, New York: Springer-Verlag, ISBN   978-1-84628-369-7
  4. Robbin, Joel W., Continuity and Uniform Continuity (PDF)
  5. "Topology of manifolds", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
  6. Donchev, Tzanko; Farkhi, Elza (1998). "Stability and Euler Approximation of One-sided Lipschitz Differential Inclusions". SIAM Journal on Control and Optimization. 36 (2): 780–796. doi:10.1137/S0363012995293694.