Talk:Lipschitz continuity

message: The definition here seems to be restricted to R. Other definitions are in higher spaces than R? is this true? http://planetmath.org/encyclopedia/LipschitzCondition.htmlUser A1 06:22, 27 September 2006 (UTC)
replies: text: The definition is not restricted to R, Lipshitz functions are defineded on any metric space, in the section on metric spaces in this article. Oleg Alexandrov (talk) 15:12, 27 September 2006 (UTC)

text: Yes, it could. --A r m y 1 9 8 7 ! ! !  12:02, 17 September 2008 (UTC)

text: Actually I see nothing in the "Real numbers" subsection which cannot apply to any other metric space (replacing |x − y| with d(x, y). I'm going to merge them, mentioning real numbers as an example. --A r m y 1 9 8 7 ! ! !  12:13, 17 September 2008 (UTC)

text: This is a usual mathematical jargon; in a definition, "X is called A if R" means "X is called A if and only if R" (just because otherwise it would not be a definition); see If and only if#More general usage. Boris Tsirelson (talk) 16:19, 26 May 2016 (UTC)

text: Nice. But you did not indicate the end of the special case. Thus I try a different design. Boris Tsirelson (talk) 19:33, 13 January 2017 (UTC)

text: Yes, “L-Lipschitz” is a standard and very common short form for “Lipschitz of constant L”, both in written and spoken mathematical language (and “L-Lip” if you want to be even shorter). In this article a short form for “Lipschitz of constant L” doesn’t seem necessary because the phrase is not repeated too many times. However, the meaning of L-Lipschitz is an information that shouldn’t be missing so I think you did well.pma 06:50, 28 September 2021 (UTC)
thread_title: The definition

message: The text currently states:
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A function f, defined on [a,b], is said to satisfy a uniform Lipschitz condition of order α > 0 on [a,b] if there exists a constant M > 0 such that
    | f(x) − f(y) | < M | x − y | ^α
for all x and y in [a,b].
---
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?
Jorn74 (talk) 22:11, 18 May 2008 (UTC)
replies: text: 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. Oleg Alexandrov (talk) 01:53, 19 May 2008 (UTC)
thread_title: uniform Lipschitz condition

message: The article metric map says that a map is metric if d_{Y}(f(x),f(y)) \leq d_{X}(x,y). This definition includes contractions, whereas the one in this article doesn't, if K needs to be "the smallest such constant". To make this definition be equivalent to the one in metric map, "K = 1" should be "K ≤ 1". (Also, depending on whether K must be "the smallest such constant" or not, the recent addition "0 < K" in the definition of contraction either has the effect of excluding constant functions from the definition, or is completely useless.) -- A r m y 1 9 8 7 ! ! !  10:21, 27 September 2008 (UTC)
thread_title:  Can ''K'' be less than 1 in short maps? 

message: The first paragraph states Lipschitz continuity condition is "stronger than regular continuity". is this true? need an example in the example section. Jackzhp (talk) 22:03, 18 May 2009 (UTC)
replies: text: It's definitely stronger. Consider x \mapsto \sqrt{x} where x \in [0,\infty); this function is continuous over its domain, but it is not Lipschitz continuous as the slope goes to infinity as x \to 0. In fact, this case is already 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 uniform continuity is the essence of the "magic" behind all of the nice features of Lipschitz continuous functions. —TedPavlic (talk) 12:39, 19 May 2009 (UTC)
thread_title: stronger?

message: 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 infinity.--Gustav Ulsh Iler (talk) 20:52, 23 October 2009 (UTC)
thread_title:  Lipschitz Constant and Examples

message: should something about Norton's Dome be added to this page?  — Preceding unsigned comment added by 72.83.192.20 (talk) 03:15, 1 November 2012 (UTC)
replies: text: Note: we now have a (very stubby) article on Norton's dome that needs to be filled out into a proper article. -- The Anome (talk) 17:51, 5 August 2014 (UTC)
thread_title:  should something about Norton's Dome be added to this page?  

message: 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 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.  — Preceding unsigned comment added by Bojeryd91 (talk • contribs) 16:46, 30 October 2014 (UTC)
thread_title:  One-sided Lipschitz 

message: 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 the graph. I'm not familiar with mathematical terminology so I'm not comfortable making a change. 69.172.155.110 (talk) 04:32, 5 June 2016 (UTC)
replies: text: No, "any" is not recommended in mathematics, since it is somewhat ambiguous: means not always "every", but also "some". Boris Tsirelson (talk) 04:58, 5 June 2016 (UTC)
thread_title:  Wording in Introduction 

message: I don't think a function is called a contraction unless it maps a metric space to itself; 0 ≤ K < 1 is not sufficient. The page should be updated to reflect this
Merudo (talk) 12:57, 17 June 2018 (UTC)
thread_title:  Contraction 

message: Am I missing something?
The section "Continuous functions that are not (globally) Lipschitz continuous" says that square root is not Lipschitz on [0,1] (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 in example 15 that The function square root is uniformly continuous on the set S = (0,infinity).  Note that zero is NOT included.
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.
Programmer in Chief (talk) 17:38, 21 January 2019 (UTC)
replies: text: Yes, the square root IS uniformly continuous on the whole [0,\infty). Just because \sqrt x \le \sqrt{x+a} \le \sqrt x + \sqrt a. Boris Tsirelson (talk) 20:13, 21 January 2019 (UTC)
thread_title:  Where is square root uniformly continuous? 

message: 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
: Continuously differentiable ⊂ Lipschitz continuous ⊂ α-Hölder continuous ⊂ uniformly continuous = continuous
On closed interval of real line, closed and bounded implies compact due to Heine–Borel theorem and uniformly continuous = continuous due to Heine–Cantor theorem? Should this be mentioned? Should the more general statement that uniform continuity implies continuity for any metric space be included? Wqwt (talk) 07:16, 4 April 2019 (UTC)
replies: text: 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. Boris Tsirelson (talk) 10:54, 4 April 2019 (UTC)
thread_title:  Inclusion chain