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The section "Continuous functions that are not (globally) Lipschitz continuous" says that square root is not
Lipschitz on (true) but that it is uniformly continuous. It seems to me that if zero is included in the interval, then it is not uniformly continuous either. The cited article [Robbin[ says
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Should the every be changed to any in the opening sentence? "here exists a definite real number such that, for every pair of points on the graph of this function". Every seems to indicate some sort of pairs that already exist, while any is a bit more clear that it can be talking about any 2 point on
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The following sentence is not clear to me: "Any such K is referred to as a
Lipschitz constant for the function Æ. The smallest constant is sometimes called the (best) Lipschitz constant; however in most cases the latter notion is less relevant." I would remove the part from "; however..." as we are
696:
I ran into this article looking for the more general "Lipschitz condition" which, unlike "Lipschitz uniform condition" or "Lipschitz continuity", does not even assume the function is differentiable everywhere. Robert Seeley's"An
Introduction to Fourier Series and Integrals", for example, defines it
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There should be more examples and in the definition it should state somewhere that in the R^2 case the inequality is simply |f(y)-f(x)| is equal to or less than K|x-y|. Also, the
Lipschitz constant should be discussed more, explicitly stating how to find it and that it cannot be less than 0 or
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Yes, âL-Lipschitzâ is a standard and very common short form for âLipschitz of constant Lâ, both in written and spoken mathematical language (and âL-Lipâ if you want to be even shorter). In this article a short form for âLipschitz of constant Lâ doesnât seem necessary because the phrase is not
1091:
The example of an one-sided
Lipschitz function does not make sense when using the definition in the section; F(x) is scalar, which makes the left-hand side of the definition vector times scalar, thus a vector, while the right-hand side is a norm of a vector. Also, the image of F(x) should be
615:
A bilipschitz function need not be either surjective or injective. However, if a bilipschitz function is bijective, its inverse function is, in fact, bilipschitz. It's even satisfied by the same constant (just switch objects with their images and rearrange the inequalities.)
347:
I've added a subtitle for Real-valued functions, to make it more easily readable and understandable, specifically by people not used to metric spaces. I'm new to
Knowledge, so unsure whether this was a good idea or if it was executed well, so feel free to expand upon it.
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Update: Sorry, I think I was missing something. Although in example 15, the set S does not include zero, when I actually read the proof I think it works even if zero is included. I don't know how to delete this comment, only update it.
635:(as the article currently implies). Furthermore, the characterization of bilipschitz function as a Lipschitz injection with Lipschitz inverse is the basic motivation for even being interested in bilipschitz mappings in the first place.
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Some people may have thought it too obvious to mention, but Iler is right that the article should point out explicitly that the
Lipschitz constant must be in between 0 and infinity. Otherwise the definition is simply incorrect.
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The definition Iler just gave is for the
Lipschitz condition, not for Lipschitz continuity. The Lipschitz condition is defined for piecewise differentiable functions, so is more general than Lipschitz continuity.
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for all x and y in . --- which appears to be the same as Hölder continuity. It also appears to be a misuse of the term `uniform', which should mean `independent of x and y', i.e. not locally
Lipschitz. Agreed?
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Yes, this is a good idea. For now, this chain "oscillates": someone replaces the last equality with strict inclusion, then someone restores it. This happens during years. A clarification would help.
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I agree that "uniform
Lipschitz condition of order α " appears the same as Hölder continuity. I think the "uniform" part in the article is right, there is nothing local in that definition.
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specified, for example to be from R^d to R^d as in "Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient" by Bouchut et al.
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the following theorem holds; which doesn't thus have to be valid for all Lipschitz continuous functions. Later in the 3rd paragraph the same condition (excluding the case
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in the examples section. Further down in the page, it also says that "Every Lipschitz continuous map is uniformly continuous, and hence a fortiori continuous." This
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I included a phrase to indicate the usage of the phrase 'L-Lipschitz' for a function that is Lipschitz continuous with Lipschitz constant L, as described here:
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The first paragraph states Lipschitz continuity condition is "stronger than regular continuity". is this true? need an example in the example section.
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382:]. Not sure if my edit was accepted but I would appreciate if someone more knowledgeable than me can confirm this usage and review the edit. Thanks,
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Every bilipschitz function is injective. A bilipschitz function is the same thing as a Lipschitz bijection whose inverse function is also Lipschitz.
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We have the following chain of strict inclusions for functions over a closed and bounded (i.e. compact) non-trivial interval of the real line
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The definition is not restricted to R, Lipshitz functions are defineded on any metric space, in the section on metric spaces in this article.
1370:? Should this be mentioned? Should the more general statement that uniform continuity implies continuity for any metric space be included?
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842:" in the definition of contraction either has the effect of excluding constant functions from the definition, or is completely useless.) --
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This definition and its link with Lipschitz continuity should be mentioned in this article, this section seems to be a good place for it.
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in example 15 that The function square root is uniformly continuous on the set S = (0,infinity). Note that zero is NOT included.
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repeated too many times. However, the meaning of L-Lipschitz is an information that shouldnât be missing so I think you did well.
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for a continuous and piecewise-differentiable function g(x) as |g(x) - g(y)| <= M|x-y| for all x and y (in the domain of g).
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This is wrong. A bilipschitz function is always injective. I hope you agree also that a bilipschitz function is surjective
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The text currently states: --- A function f, defined on , is said to satisfy a uniform Lipschitz condition of order α : -->
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No, "any" is not recommended in mathematics, since it is somewhat ambiguous: means not always "every", but also "some".
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In the first paragraph of the definition section it is said, that the function f is called Lipschitz continuous
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The definition here seems to be restricted to R. Other definitions are in higher spaces than R? is this true?
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continuity is the essence of the "magic" behind all of the nice features of Lipschitz continuous functions. â
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the graph. I'm not familiar with mathematical terminology so I'm not comfortable making a change.
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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Hi, there is also writen that we assume that alpha is between zero and one, then it should be
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I don't think a function is called a contraction unless it maps a metric space to itself; 0 â€
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Nice. But you did not indicate the end of the special case. Thus I try a different design.
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But this is not how bilipschits functions were defined in the article. The definition was:
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that is Lipschitz and has a Lipschitz inverse, then it is trivially injective. Ideas?
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This definition includes contractions, whereas the one in this article doesn't, if
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Could it be useful to talk about the familiar metric space R^n as a special case?
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must be "the smallest such constant" or not, the recent addition "0 <
1164:< 1 is not sufficient. The page should be updated to reflect this
249:). I'm going to merge them, mentioning real numbers as an example. --
580:{\displaystyle {\frac {1}{K}}d(x,y)\leq d'(f(x),f(y))\leq Kd(x,y)}
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should something about Norton's Dome be added to this page?
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should something about Norton's Dome be added to this page?
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Yes, the square root IS uniformly continuous on the whole
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http://planetmath.org/encyclopedia/LipschitzCondition.html
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In other words, if we define a bilipschitz function as a
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that needs to be filled out into a proper article. --
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109:and see a list of open tasks.
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1238:{\displaystyle [0,\infty ).}
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976:. In fact, this case is
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