Lipschitz continuity

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 definite 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; this bound is called a "Lipschitz constant" of the function (or "modulus of uniform continuity"). For instance, every function that has bounded first derivatives is Lipschitz.^[1]

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 inclusions for functions over a closed and bounded^[2] subset of the real line

Continuously differentiable ⊆ Lipschitz continuous ⊆ α-Hölder continuous ⊆ uniformly continuous ⊆ continuous

where 0 < α ≤1. We also have

Lipschitz continuous ⊆ absolutely continuous ⊆ bounded variation ⊆ differentiable almost everywhere

Definitions

For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph, so that the graph always remains entirely outside 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(y₁, y₂) = |y₁ − y₂|, and X might be a subset of R), 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,

d_Y(f(x_1), f(x_2)) \le K d_X(x_1, x_2).

^[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 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,

\frac{d_Y(f(x_1),f(x_2))}{d_X(x_1,x_2)}\le K.

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

d_Y(f(x), f(y)) \leq 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 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 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. 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 f(x) = |x|

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] as well as 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) where 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.

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

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 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 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 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 extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map f : U → R^m, where U is an open set in Rⁿ, is almost everywhere differentiable. Moreover, if K is the best Lipschitz constant of f, then $\|Df(x)\|\le K$ whenever the total derivative Df exists.
For a differentiable Lipschitz map f : U → R^m the inequality $\|Df\|_{\infty,U}\le K$ holds for the best Lipschitz constant of f, and it turns out to be an equality if the domain U is convex.
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 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 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, because every polynomial is Lipschitz continuous).
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).
For a family of Lipschitz continuous functions f_α with common constant, the function $\sup_\alpha f_\alpha$ (and $\inf_\alpha f_\alpha$ ) 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). An extension is provided by

\tilde f(x):=\inf_{u\in U}\{ f(u)+k\, d(x,u)\},

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 : 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;^[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

(x_1-x_2)^T(F(x_1)-F(x_2))\leq C\Vert x_1-x_2\Vert^2

for some C 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

\begin{cases} F:\mathbf{R}^2\to\mathbf{R},\\ F(x,y)=-50(y-\cos(x)) \end{cases}

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.

References

↑ Sohrab, H. H. (2003). Basic real analysis (Vol. 231). Birkhäuser
↑ Compactness
↑ Searcóid, Mícheál Ó (2006), Metric spaces, Springer undergraduate mathematics series, Berlin, New York: Springer-Verlag, ISBN 978-1-84628-369-7 , section 9.4
↑ Robbin, Joel W., Continuity and Uniform Continuity (PDF)
↑ SpringerLink: Topology of manifolds
↑ 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.

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