Talk:Singularity (mathematics)

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The example of the absolute value function having a singularity at x=0 might not be the best, because the absolute value function is not complex differentiable anywhere.

I don't understand the sentence: The algebraic set defined by y2 = x also has a singularity at (0,0), this time because it has a "corner" at that point.

As a set in the plan (i.e., as an imbedded 1-manifold) it is perfectly well behaved. Thought of as a (multi-valued) function it has an infinite slope.

Per haps "... because it has a vertical tangent at that point."

Maybe I'm missing something, which is why I didn't just make the edit myself.... Better not to be bold and confused at the same time. ;-> Jeff 01:40 Apr 10, 2003 (UTC)

Yeah, I deleted the y2=x example, because it's just a sideways parabola. It doesn't have any undefined areas, or anything else that could be considered singularities. If you count the negative x values, maybe, but that wasn't included in the example.
There's other bad examples. Like y=|x|. You can take the absolute value of zero, right? Isn't it just zero? — Preceding unsigned comment added by (talk) 17:03, 20 May 2017 (UTC)

Should we add a fourth kind of singularity in complex analysis -- a branch point? Michael Hardy 02:43 Apr 10, 2003 (UTC)

Are singularities in lower dimensions NOT SO IN THE HIGHER DIMENSIONS?This can be proved in this manner-suppose a curve is given ,then write it's parametric equation.treat the parameter as the new dimension .trace the curve in this new coordinate system .sometimes we see the singularity has vanished in the higher dimension.[user:Preetam ]

Shouldn't this page be merged with Singularity theory ? --Piotr Konieczny aka Prokonsul Piotrus 21:37, 12 Jul 2004 (UTC)

Shouldn't this be at singularity (mathematics)? - Fredrik | talk 19:01, 12 Apr 2005 (UTC)

It would be better to have that as disambiguation, I think. Singularity of a function would be one alternate name. There is also singularity of a differential equation to think about. Charles Matthews 19:14, 12 Apr 2005 (UTC)

We seem to bw missing singular matricies (determinant zero). I'd like to add a section about one singularity theory in more details (just 1 paragraph). Also should we include a disambig link to Singularity? --Salix alba (talk) 14:36, 24 January 2006 (UTC)


A lot of these seem to be closely linked.

Start with a function f : Rm -> Rn. The first derivative dfˈis a linear map or an m by n matrix. f will be singular (in terms of algebraic geometry/singularity theory) when the matrix drops rank. (i.e. is singular).

The commutative algebra ideal's are generilisations of this concept, (think of the ideal generated by the derivatives).

Likewise the singular solutions of ordinary differential equations are simarly linked. --Salix alba (talk) 11:53, 25 January 2006 (UTC)

This may not belong here, but it's the first place that one can add comments to this page - in any case, is "ejaculate" the correct mathematical term for a function that becomes unbounded as it approaches some value (the example on this page being f(x) = 1/x as x > 0)? Is this some sort of vandalism that someone should consider correcting? jmdeur 14:30, 21 March 2008 (UTC)

Number of Types of Singularities in Complex Analysis[edit]

There are more than four types of singularities in Complex Analysis. I will correct this mistake if there is no disagreement on this issue. Furthermore, ranking them is appropriate, as the differences are significant. Tparameter 05:53, 9 December 2006 (UTC)


I (physicist) don't understand what U \ {a} means. Perhaps somebody who does could add a link to something which explains it? Or make it more explicit/obvious? It's in the Complex Analysis section. Thanks -- (talk) 18:22, 29 July 2009 (UTC)

It means the set U without the element 'a.' (though, 'a' need not be a member of the set U. In which case U\{a} = U.) Got it? I don't think we should change it. It's very common math notation. futurebird (talk) 07:38, 20 September 2010 (UTC)

Etymology: What is the "single" that "singularity" refers to?[edit]

Could someone explain what it is that is single that prompts the name "singularity? Gwideman (talk) 00:21, 20 September 2010 (UTC)

if you take the second definition of singular Being the only one of the kind; unique. It most common for a singularity to occur at an isolated point, 1/x has a unique point at 0, which has different behaviour to all other points, hence is singular or a singularity.--Salix (talk): 06:48, 20 September 2010 (UTC)
I have never thought of this before but it makes a lot of sense. I wonder if we can source it? futurebird (talk) 07:39, 20 September 2010 (UTC)
Thanks, Salix, for the reply. I had indeed speculated about that being the origin of the term, but it seems a little indirect -- especially since a function might well have more than a single singularity. So like Futurebird I'm wondering if there's an authoritative source for that account? Gwideman (talk) 21:54, 25 September 2010 (UTC)

Meaning of the word[edit]


The function g(x) = |x| (see absolute value) also has a singularity at x = 0, since it is not differentiable there. Similarly, the graph defined by y2 = x also has a singularity at (0,0), this time because it has a "corner" (vertical tangent) at that point.

As the section on real analysis says: A point of continuity, which is not a singularity, is a value of c for which f(c − ) = f(c) = f(c + ). Why is the intro using a different defn for the meaning of the word, and where does it get it from?

seems odd to me. I think |x| has no singularities, but its derivative has one at 0. Ditto, y2=x. And calling a vertical tangent a corner is odd, too William M. Connolley (talk) 11:02, 31 December 2010 (UTC)

Singularity and discontinuity are not the same[edit]

I believe this voice is flawed. It keeps on confusing two concepts which should not be confused, at least here! Singularities and discontinuities are not the same: though it may be true that every discontinuity is alaso singular point (depending of the concept of "well-behavedness" you have), the converse certainly is not. y=1/x has a singular point at {0}, but it's a continuos funtion - hence it can't have any discontinuities. The fact is that discontinuity is a domain-related concept: it has no sense to discuss about continuity in points which not belong to the domain of the given function. On the contrary, singularity is a concept always referring to a broader set than the domain (with the limit case of this broader set = domain): a continuos function may be singular in some extension of its domain (again, y=1/x is continuos in its domain R-{0}, but singular in R}. I think the entry should be fixed, and made coherent with the content of which seems to be more precise. — Preceding unsigned comment added by (talk) 16:28, 11 July 2012 (UTC)

I agree (talk) 20:46, 4 April 2017 (UTC)

Change to Singularity (mathematics)[edit] (talk) 17:46, 14 March 2014 (UTC)

Singularity (mathematics) exists already as a redirect to this article. I agree with moving Mathematical singularity to Singularity (mathematics), because "mathematical singularity" is a terminology which is rarely used. However, this move can only be done by an administrator and needs thus a request for move. I'll support such a request. D.Lazard (talk) 19:57, 14 March 2014 (UTC)

Requested move[edit]

The following discussion is an archived discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the proposal was moved. --BDD (talk) 17:02, 26 March 2014 (UTC)

Mathematical singularitySingularity (mathematics) – Esoteric term. See above. (talk) 23:04, 16 March 2014 (UTC) (talk) 23:04, 16 March 2014 (UTC)

  • Oppose WP:NATURALDIS prefer using natural disambiguation over parenthetical disambiguation. There are many hits on Google books and Google for "mathematical singularity"; and even Google scholar. So "mathematical singularity" is used in technical works as well. Especially in research that are not solely a mathematics paper, but instead is some other field of science. -- (talk) 06:35, 17 March 2014 (UTC)
  • Support In mathematics, singularity is a technical term, by its own, without the adjective mathematical. When this adjective is used, this appears to be in physics, when classifying the physical singularities, which may or not be explained by mathematical singularities (all the hits in the first page of Google Scholar, for "mathematical singularity" are physical articles). Thus "mathematical singularity" is far less common than "singularity" alone. Mathematical group and mathematical ring are redirects to group (mathematics) and ring (mathematics). The same must occur for "Singularity". D.Lazard (talk) 09:24, 17 March 2014 (UTC)
  • The google book results are quite odd. The first three are flaky: one on the Singularity Hypotheses the second "A Critique of Pure Physics" looks equally leftfield, and the third is on science fiction. Next we have eight applied science books. The first genuine mathematical books is "Arnol'ds "The Theory of Singularities and Its Applications" with the matching line being "in which the theorems of mathematical singularity theory provide" so not really a match. On the whole first page of 100 title I can count maybe three mathematical works using the term. Applied scientist can be forgiven for getting the nomeculture wrong but its not a term really used in mathematics.
The other concern I have is that there are two quite distinct meanings on singularity in mathematics, A) points where a function has a discontinuity and B) and singularities in the context of Singularity theory. The article is very much weighted towards the first. Is there a way which this distinction could be made clearer in the article title.--Salix alba (talk): 10:43, 17 March 2014 (UTC)
About the "other concern" of Salix alba: IMO, all notions of singularity in mathematics are related to points where some function or its derivative becomes undefined: a curve or a surface defined by an differentiable implicit equation has a singularity if the normal vector is not defined. Nevertheless, it is difficult to give a general definition of "singularity" which covers all the cases. A similar difficulty occurs with the related definition of critical point (mathematics). I have recently edited critical point (mathematics) in a tentative to fix this point. My opinion is that a similar work is needed for both mathematical singularity and Singularity theory. D.Lazard (talk) 11:40, 17 March 2014 (UTC)
  • Support per WP:COMMONNAME, and D.Lazard (talk · contribs). The current title suggests that "mathematical singularity" is the term commonly used in reliable sources to refer to the topic of this article. That's wrong and misleading. This proposal will correct that. --B2C 01:09, 26 March 2014 (UTC)
The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.