Show Sqrt X is Continuous but Not Lipschitz Metric Space

For a Lipschitz continuous serve as, there exists a double cone (white) whose beginning can be moved along the graph in order that the entire graph at all times stays out of doors the double cone

In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a sturdy form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real quantity such that, for each and every pair of issues on the graph of this function, the absolute value of the slope of the road connecting them is not more than this real number; the smallest such bound is named the Lipschitz consistent of the serve as (or modulus of uniform continuity). For example, each and every serve as that has bounded first derivatives is Lipschitz continuous.^[1]

In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and specialty of the strategy to an initial value problem. A different type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem.^[2]

We have the next chain of strict inclusions for functions over a closed and bounded non-trivial period of the real line

Continuously differentiable ⊂ Lipschitz continuous ⊂ $\displaystyle \alpha$ -Hölder continuous

the place $\displaystyle 0<\alpha \leq 1$ . We even have

Lipschitz continuous ⊂ absolutely continuous ⊂ uniformly continuous.

Definitions [edit]

Given two metric spaces (X, d _X ) and (Y, d _Y ), the place d _X denotes the metric at the set X and d _Y is the metric on set Y, a function f : X → Y is called Lipschitz steady if there exists an actual constant Ok ≥ 0 such that, for all x ₁ and x ₂ in X,

$\displaystyle d_Y(f(x_1),f(x_2))\leq Kd_X(x_1,x_2).$ ^[3]

Any such K is referred to as a Lipschitz constant for the function f and f may also be referred to as Okay-Lipschitz. The smallest consistent is often referred to as the (perfect) Lipschitz constant ^[4] of f or the dilation or dilatation ^{[5] : p. 9, Definition 1.4.1 [6] [7]} of f. If K = 1 the function is known as a short map, and if 0 ≤ Okay < 1 and f maps a metric area to itself, the serve as is known as a contraction.

In particular, a real-valued function f : R → R is called Lipschitz continuous if there exists a positive real consistent Ok such that, for all real x ₁ and x ₂,

$\leq Okay$

In this example, Y is the set of real numbers R with the usual metric d _Y (y₁ , y₂ ) = |y₁ − y₂ |, and X is a subset of R.

In normal, the inequality is (trivially) glad if x ₁ = x ₂. Otherwise, one can equivalently outline a serve as to be Lipschitz continuous if and only if there exists a continuing K ≥ 0 such that, for all x ₁ ≠ x ₂,

$\displaystyle \frac d_Y(f(x_1),f(x_2))d_X(x_1,x_2)\leq Okay.$

For real-valued functions of several actual variables, this holds if and only if absolutely the worth of the slopes of all secant lines are bounded by means of Okay. The set of traces of slope Okay passing via some extent at the graph of the serve as bureaucracy a circular cone, and a function is Lipschitz if and only if the graph of the function everywhere lies totally outside of this cone (see determine).

A serve as is known as in the community Lipschitz continuous if for each and every x in X there exists a neighborhood U of x such that f limited to U is Lipschitz continuous. Equivalently, if X is a locally compact metric area, then f is in the community Lipschitz if and best if it is Lipschitz steady on each and every compact subset of X. In spaces that aren't locally compact, it is a vital however not a sufficient situation.

More usually, a serve as f defined on X is claimed to be Hölder steady or to satisfy a Hölder condition of order α > Zero on X if there exists a continuing M ≥ 0 such that

$\displaystyle d_Y(f(x),f(y))\leq Md_X(x,y)^\alpha$

for all x and y in X. Sometimes a Hölder condition of order α is often known as a uniform Lipschitz situation of order α > 0.

For a real number K ≥ 1, if

$\displaystyle \frac 1Okd_X(x_1,x_2)\leq d_Y(f(x_1),f(x_2))\leq Kd_X(x_1,x_2)\quad \textual content for all x_1,x_2\in X,$

then f is known as Ok-bilipschitz (additionally written Okay-bi-Lipschitz). We say f is bilipschitz or bi-Lipschitz to mean there exists one of these Ok. A bilipschitz mapping is injective, and is if truth be told a homeomorphism onto its image. A bilipschitz serve as is the same factor as an injective Lipschitz serve as whose inverse function could also be Lipschitz.

Examples [edit]

Lipschitz continuous functions

• The serve as $\displaystyle f(x)=\sqrt x^2+5$ defined for all real numbers is Lipschitz continuous with the Lipschitz consistent Ok = 1, because it is everywhere differentiable and absolutely the value of the spinoff is bounded above by way of 1. See the first property indexed under beneath "Properties".
• Likewise, the sine serve as is Lipschitz steady as a result of its by-product, the cosine function, is bounded above by means of 1 in absolute price.
• The serve as f(x) = |x| outlined at the reals is Lipschitz continuous with the Lipschitz constant equal to 1, via the reverse triangle inequality. This is an instance of a Lipschitz continuous function that is not differentiable. More typically, a norm on a vector space is Lipschitz steady with admire to the related metric, with the Lipschitz consistent equal to 1.

Lipschitz steady functions that are not all over differentiable

• The function $\displaystyle f(x)=$

Lipschitz continuous purposes which are all over the place differentiable however not incessantly differentiable

Continuous functions that aren't (globally) Lipschitz steady

• The function f(x) =√ x defined on [0, 1] is now not Lipschitz continuous. This function becomes infinitely steep as x approaches Zero since its spinoff becomes endless. However, it is uniformly steady,^[8] and both Hölder continuous of sophistication C ^{0, α} for α ≤ 1/2 and likewise absolutely continuous on [0, 1] (either one of which suggest the previous).

Differentiable purposes that are not (in the neighborhood) Lipschitz steady

• The function f outlined by f(0) = Zero and f(x) =x ^3/2sin(1/x) for 0<x≤1 provides an example of a function this is differentiable on a compact set whilst now not in the community Lipschitz as a result of its by-product serve as isn't bounded. See also the first belongings beneath.

Analytic purposes that don't seem to be (globally) Lipschitz continuous

• The exponential function becomes arbitrarily steep as x → ∞, and therefore is now not globally Lipschitz steady, despite being an analytic function.
• The function f(x) =x ² with area all real numbers is no longer Lipschitz continuous. This serve as turns into arbitrarily steep as x approaches infinity. It is on the other hand locally Lipschitz continuous.

Properties [edit]

• An all over differentiable function g :R →R is Lipschitz steady (with Okay = sup |g′(x)|) if and only if it has bounded first derivative; one direction follows from the mean value theorem. In specific, any frequently differentiable serve as is locally Lipschitz, as steady functions are in the community bounded so its gradient is in the neighborhood bounded as well.
• A Lipschitz function g :R →R is absolutely continuous and due to this fact is differentiable almost everywhere, this is, differentiable at each and every level outside a collection of Lebesgue measure 0. Its derivative is essentially bounded in magnitude by means of the Lipschitz consistent, and for a < b, the difference g(b) −g(a) is the same as the integral of the derivative g′ at the period [a,b].
  • Conversely, if f : I → R is absolutely steady and thus differentiable almost everywhere, and satisfies |f′(x)| ≤ Okay for almost all x in I, then f is Lipschitz steady with Lipschitz consistent at most Ok.
  • More in most cases, Rademacher's theorem extends the differentiability outcome to Lipschitz mappings between Euclidean spaces: a Lipschitz map f :U →R ^m , the place U is an open set in R ⁿ , is almost everywhere differentiable. Moreover, if K is the best Lipschitz constant of f, then $\leq K$ on every occasion the total derivative Df exists.
• For a differentiable Lipschitz map $\displaystyle f:U\to \mathbb R ^m$ the inequality $\displaystyle \$ holds for the best Lipschitz consistent $\displaystyle K$ of $\displaystyle f$ . If the area $\displaystyle U$ is convex then in truth $Df\$ .^{[ further explanation needed ]}
• Suppose that f_n is a sequence of Lipschitz continuous mappings between two metric areas, and that every one f_n have Lipschitz constant bounded by way of some Okay. If f_n converges to a mapping f uniformly, then f could also be Lipschitz, with Lipschitz consistent bounded by the same K. In explicit, this means that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz consistent is a closed and convex subset of the Banach space of constant functions. This outcome does not dangle for sequences through which the purposes will have unbounded Lipschitz constants, on the other hand. In fact, the distance of all Lipschitz purposes on a compact metric area is a subalgebra of the Banach area of constant purposes, and thus dense in it, an fundamental consequence of the Stone–Weierstrass theorem (or as a consequence of Weierstrass approximation theorem, as a result of each and every polynomial is in the community Lipschitz steady).
• Every Lipschitz steady map is uniformly continuous, and therefore a fortiori continuous. More most often, a suite of functions with bounded Lipschitz constant bureaucracy an equicontinuous set. The Arzelà–Ascoli theorem implies that if f_n is a uniformly bounded collection of functions with bounded Lipschitz consistent, then it has a convergent subsequence. By the results of the previous paragraph, the restrict function could also be Lipschitz, with the similar sure for the Lipschitz consistent. In particular the set of all actual-valued Lipschitz purposes on a compact metric area X having Lipschitz consistent ≤K  is a locally compact convex subset of the Banach house C(X).
• For a family of Lipschitz continuous functions f _α with commonplace consistent, the serve as $\displaystyle \sup _\alpha f_\alpha$ (and $\displaystyle \inf _\alpha f_\alpha$ ) is Lipschitz continuous as well, with the similar Lipschitz constant, provided it assumes a finite price no less than at some degree.
• If U is a subset of the metric space M and f : U → R is a Lipschitz steady serve as, there all the time exist Lipschitz steady maps M → R which lengthen f and feature the same Lipschitz constant as f (see additionally Kirszbraun theorem). An extension is supplied by means of

$\displaystyle \tilde f(x):=\inf _u\in U\f(u)+k\,d(x,u)\,$

where k is a Lipschitz constant for f on U.

Lipschitz manifolds [edit]

A Lipschitz structure on a topological manifold is outlined the use of an atlas of charts whose transition maps are bilipschitz; that is conceivable because bilipschitz maps form a pseudogroup. Such a construction permits one to define in the neighborhood Lipschitz maps between such manifolds, in a similar way to how one defines clean maps between smooth manifolds: if M and N are Lipschitz manifolds, then a function $\displaystyle f:M\to N$ is in the neighborhood Lipschitz if and only if for each pair of coordinate charts $\displaystyle \phi :U\to M$ and $\displaystyle \psi :V\to N$ , the place U and V are open units within the corresponding Euclidean areas, the composition

$$\displaystyle \psi ^-1\circ f\circ \phi :U\cap (f\circ \phi )^-1(\psi (V))\to N$$

is in the community Lipschitz. This definition does now not depend on defining a metric on M or N.^[9]

This structure is intermediate between that of a piecewise-linear manifold and a topological manifold: a PL construction provides upward push to a novel Lipschitz structure.^[10] While Lipschitz manifolds are carefully associated with topological manifolds, Rademacher's theorem allows one to do research, yielding various applications.^[9]

One-sided Lipschitz [edit]

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^[11] if

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

for some C and for all x ₁ and x ₂.

It is imaginable that the function F will have an overly large Lipschitz consistent but a fairly sized, and even negative, one-sided Lipschitz constant. For example, the serve as

$\displaystyle \begincasesF:\mathbf R ^2\to \mathbf R ,\\F(x,y)=-50(y-\cos(x))\finishcases$

has Lipschitz constant Okay = 50 and a one-sided Lipschitz consistent C = 0. An instance which is one-sided Lipschitz but no longer Lipschitz continuous is F(x) = e ^−x , with C = 0.

See also [edit]

References [edit]

1. ^ Sohrab, H. H. (2003). Basic Real Analysis. Vol. 231. Birkhäuser. p. 142. ISBN0-8176-4211-0.
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 collection, Berlin, New York: Springer-Verlag, ISBN978-1-84628-369-7
4. ^ Benyamini, Yoav; Lindenstrauss, Joram (2000). Geometric Nonlinear Functional Analysis. American Mathematical Society. p. 11. ISBN0-8218-0835-4.
5. ^ Burago, Dmitri; Burago, Yuri; Ivanov, Sergei (2001). A Course in Metric Geometry. American Mathematical Society. ISBN0-8218-2129-6.
6. ^ Mahroo, Omar A; Shalchi, Zaid; Hammond, Christopher J (2014). "'Dilatation' and 'dilation': trends in use on both sides of the Atlantic". British Journal of Ophthalmology. 98 (6): 845–846. doi:10.1136/bjophthalmol-2014-304986.
7. ^ Gromov, Mikhael (1999). "Quantitative Homotopy Theory". In Rossi, Hugo (ed.). Prospects in Mathematics: Invited Talks on the Occasion of the 250th Anniversary of Princeton University, March 17-21, 1996, Princeton University. American Mathematical Society. p. 46. ISBN0-8218-0975-X.
8. ^ Robbin, Joel W., Continuity and Uniform Continuity (PDF)
9. ^ ^{a b} Rosenberg, Jonathan (1988). "Applications of analysis on Lipschitz manifolds". Miniconferences on harmonic research and operator algebras (Canberra, 1987). Canberra: Australian National University. pp. 269–283. MR954004
10. ^ "Topology of manifolds", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
11. ^ 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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