View source for Talk:Continuous function

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== Counterexample for the equivalence of two definitions for topological spaces ==

The article originally stated:

{{Quote frame|
* A function <math>f:X\rightarrow Y</math> is continuous at a point <math>x\in X</math> if and only if for any neighborhood of its image <math>f(x)\in Y</math> the preimage is again a neighborhood of <math>x\in X</math>.
This definition can be shown to be equivalent to the same statement, with open neighborhoods instead of neighborhoods.}}

The same statement with open neighborhoods instead of neighborhoods would be:

{{Quote frame|
* A function <math>f:X\rightarrow Y</math> is continuous at a point <math>x\in X</math> if and only if for any open neighborhood of its image <math>f(x)\in Y</math> the preimage is again an open neighborhood of <math>x\in X</math>.}}

The last sentence then say these two statements are equivalent. That is false, here is a counterexample:

Let <math>(X,\mathcal{T}_X)</math> and <math>(Y,\mathcal{T}_{Y})</math> be two topological spaces with <math>X = \{a,b,c\}</math>, <math>\mathcal{T}_{X} = \{\emptyset, \{a,b,c\}, \{a\}\}</math>, <math>Y = \{w,y,z\}</math> and <math>\mathcal{T}_{Y} = \{\emptyset, \{w,y,z\}, \{w,y\}\}</math>.

Let <math>f \colon X \rightarrow Y</math> be a function from the first topological space to the second defined by <math>f(x) = \begin{cases}
  w, & \text{if } x = a \\
  y, & \text{if } x = b \\
  z, & \text{if } x = c
\end{cases}</math>.

The only two neighborhoods of <math>f(a)</math> are <math>\{w,y\}</math> and <math>\{w,y,z\}</math>.
<math>f^{-1}(\{w,y\}) = \{a,b\}</math> and <math>f^{-1}(\{w,y,z\}) = \{a,b,c\}</math>. Both <math>\{a,b\}</math> and <math>\{a,b,c\}</math> are neighborhoods of <math>a</math> so according to the first definition the function <math>f</math> is continuous at <math>a</math>.

Now, the only two open neighborhoods of <math>f(a)</math> are also <math>\{w,y\}</math> and <math>\{w,y,z\}</math>.
<math>f^{-1}(\{w,y\}) = \{a,b\}</math> and <math>\{a,b\}</math> is not an open neighborhoods of <math>a</math>. Therefore, according to the second definition the function <math>f</math> is NOT continuous at <math>a</math>.

Therefore, the two definitions can't be equivalent. Otherwise, the function <math>f</math> would be continuous under the two definitions. Hence, I have deleted the sentence claiming that the two definitions are equivalent.

[[User:Pyrrhonist05|Pyrrhonist05]] ([[User talk:Pyrrhonist05|talk]]) 17:42, 5 November 2017 (UTC)

:True, the relevant neighborhoods of ''a'' need not be open, but relevant neighborhoods of ''f''(''a'') may belong (WLOG) to any basis of the filter, in particular, be open. Thanks to [[User:Pyrrhonist05|Pyrrhonist05]]. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 21:12, 5 November 2017 (UTC)

I've moved this from the article.  I had copy edited something similar out of the article several months ago.  This appears to restate the definition trivially in several ways, but I don't think it makes the definition any easier to understand:
{{quote frame| This definition can also be written symbolically as <math>\forall N \in N(f(x)),\ f^{-1}(N) \in N(x)</math> where <math>N(x)</math> is the [[neighborhood system]] of <math>x</math>.

Since neighborhoods systems are [[Filter_(mathematics)|filter]]s, they are [[upper set]]s. Therefore it is sufficient to say that for a function <math>f</math> to be continuous at a point <math>x</math> there must exist, for every neighborhood <math>N_{f(x)}</math> of <math>f(x)</math>, a neighborhood <math>N_x</math> of <math>x</math> ''contained'' in the preimage <math>f^{-1}(N_{f(x)})</math>. Symbolically, it must be true that <math>\forall N_{f(x)} \in N(f(x)),\ \exists N_x \in N(x) : N_x \subseteq f^{-1}(N_{f(x)})</math>.

In this second definition, it makes no difference if neighborhoods are restricted to open neighborhoods only. That is, it is equivalent to require that for every ''open'' neighborhood <math>U_{f(x)}</math> of <math>f(x)</math>, there exist an ''open'' neighborhood <math>U_x</math> of <math>x</math> contained in the preimage <math>f^{-1}(U_{f(x)})</math>.

These last two definitions can also be restated to use images of the function <math>f</math> rather than the preimages. That is, one may instead say that the function <math>f</math> is continuous at <math>x</math> if for every neighborhoods <math>N_{f(x)}</math> of <math>f(x)</math>, there exist a neighborhood <math>N_x</math> of <math>x</math> such that its image <math>f(N_x)</math> is contained in <math>N_{f(x)}</math>. Again, restricting neighborhoods to open neighborhoods only doesn't make a difference.}}
Also, we usually don't state things in the notation of symbolic logic on Wikipedia (note that we don't do this for the other definitions of continuity).  So, I've removed it pending discussion.  [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 14:11, 6 November 2017 (UTC)
:
: I didn't reintroduced this paragraph because it made the definition clearer but because I think it gave valuable information. I agree with you that it don't help to understand the first definition but I find that the definitions it adds are not merely trivial restatements of the first one. If you already know what are neighborhoods systems and the fact that they are filters (and so upper sets), it's true that the definitions are not really involved. But if you don't really know about neighborhoods systems being filters I think it is not so clear that the definition can be restated this way and that they are equivalent. Also, the first time I stumbled across this section of the page, I didn't known neighborhoods systems, filter and upper sets and it makes me discover these concepts. Therefore, I find that there is an opportunities in this section to introduce theses concept to the reader. Additionally, you may say that because all theses definitions are all equivalent we could just give one and forgot about the others but as a mathematician you know that sometimes when we try to prove something we don't succeed by using the first definition but then by using only a subtly changed definitions we manage to do it. What I want to convey is that I think it is important to give several definitions, at least to let people know there isn't only one.  At the same time, I'm not under the impression that this content clutter the section or make it less readable so I don't see any reason for not keeping it. As for the symbolic notation, we could just remove it if that is a problem. [[User:Pyrrhonist05|Pyrrhonist05]] ([[User talk:Pyrrhonist05|talk]]) 18:36, 7 November 2017 (UTC)
:
: {{Reply to|Sławomir Biały|D.Lazard}} Any thoughts on this ?

:: To avoid being guilty of [[wp:original research]] here, it would help to have a source. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 10:08, 8 November 2017 (UTC)
:: I agree with Sławomir that this text is not useful for understanding the definition. It is also unnecessarily too [[WP:TECHNICAL]], by using terms (filter and upper set), which are unknown by most users that know of continuous functions. Also, the last sentence consists of replacing a sentence like "this object has this property" by "there is an object having this property". This is never useful, and is confusing here.
:: By the way, this section is not an alternative definition, but the standard definition of the continuity at a point. I'll thus move and rename this section. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:40, 8 November 2017 (UTC)
::: I never said that the content I reintroduced helped to understand the first definition, I said that it added valuable information. In my opinion, the definitions added are not the same as the first and are not merely simple restatement of it either. As for [[WP:TECHNICAL]] I think there is point where we just can't avoid to be technical. Should we remove the definitions using the hyperreals or oscillation because  it is too technical ? Off course no ! If someone doesn't know filters and upper sets, I think it is fine since he or she can still understand and use the definitions and understand what is continuity without them. [[User:Pyrrhonist05|Pyrrhonist05]] ([[User talk:Pyrrhonist05|talk]]) 15:24, 8 November 2017 (UTC)

== Image in section "Continuous functions between topological spaces" is right ==

The image is reproduced here for clarity and completeness:
[[File:continuity topology.svg|center|frame|Continuity of a function at a point.]]
This image is referring to this definition of continuity at a point between two topological spaces:
{{Quote frame|
* A function <math>f:X \rightarrow Y</math> is continuous at <math>x \in X</math> if for each neighborhood <math>V</math> of <math>f(x)</math> there exists a neighborhood <math>U</math> of <math>x</math> such that <math>f(U) \subseteq V</math>.}}
This definition is far from being unusual, I googled "topological space continuous function" and I found these web pages and PDF on the first result page:
* [https://www.math.ucdavis.edu/~hunter/book/ch4.pdf], page 4 definition 4.6.
* [https://wolfweb.unr.edu/homepage/jabuka/Classes/2009_spring/topology/Notes/03%20-%20Continuous%20functions.pdf], page 1 definition 1.1.
* [http://mathstat.slu.edu/~bart/F07-mt531/Section18Cont-Fun.pdf], page 1 Theorem 18.1, 4th item.
* [http://www.emathzone.com/tutorials/general-topology/continuity-in-topological-spaces.html], first paragraph
Now, one could argue that they are self-published works, so here a (more difficult to find) published work mentioning this definition:
* {{cite book |last=Willard |first=Stephen |date=2012 |title=General Topology |url=https://books.google.fr/books?id=UrsHbOjiR8QC&lpg=PP1&dq=topology%20continuous%20function&hl=fr&pg=PA44#v=onepage&q=topology%20continuous%20function&f=false |publisher=Courier Corporation |page=44 |isbn=9780486131788 }}
Also <math>U</math> is not ''chosen'' after <math>V</math>, a <math>U</math> such that <math>f(U) \subseteq V</math> should just ''exists'' for (every) <math>V</math> (neighborhoods) and the image just show that by picturing one such <math>U</math> and illustrating what <math>f(U) \subseteq V</math> means . One could say that the picture doesn't shows ''every'' <math>V</math> (neighborhoods) but it's impossible to put everything in an image. An image is here to illustrate something so that it is better understood by schematizing things. Therefore the image is correct and should be restored because an image is always welcome to illustrate a point. [[User:Pyrrhonist05|Pyrrhonist05]] ([[User talk:Pyrrhonist05|talk]]) 18:29, 8 November 2017 (UTC)

== Incorrect definition of ''G''<sup>k</sup> continuity ==

The wiki claims that these are identical to their curvature continuity (''C''<sup>k</sup>) counterparts, but they are not. The geometric continuity (''G''<sup>k</sup>) conditions require continuity of the specified degree along their parameterization in addition to the continuity in the space in which they are embedded. See p. 185ff of Gerald Farin's "Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide", 2nd ed. for details.

:You are right. Thank you. Now corrected. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 06:23, 19 January 2019 (UTC)

== On the "Weierstrass and Jordan definition" ==

Shouldn't this be "uniformly continuous", and not just "continuous"? There is a discrepancy here between this page and the [[https://en.wikipedia.org/wiki/Uniform_continuity#Definition_for_functions_on_metric_spaces| uniform continuity page]]. On this page (continuous function), it is simply called "continuous," but on the other (uniform continuity), it says "The difference between uniform continuity, versus ordinary continuity at every point, is that in uniform continuity the value of δ depends only on ε and not on the point in the domain. " I think something should be cleared up, here.  <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/2603:9001:2902:1300:849B:38FF:720A:4408|2603:9001:2902:1300:849B:38FF:720A:4408]] ([[User talk:2603:9001:2902:1300:849B:38FF:720A:4408#top|talk]]) 05:36, 24 September 2019 (UTC)</small> <!--Autosigned by SineBot-->
:There is no discrepancy, as, here, <math>x_0</math> is defined before <math>\delta,</math> which implies that <math>\delta</math> may depend on <math>x_0.</math> [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 06:36, 24 September 2019 (UTC)

==Definition in terms of limits of functions==
I like this definition because its somewhat easier to check against, as the example of the signum function demonstrates. It also looks very similar to the definition of Scott continuity - in fact Scott continuity looks like a generalization of this definition of continuity. If this is in fact the case, could we state this connection somewhere? I imagine many people from a programming semantics background would be interested! [[User:Houseofwealth|Houseofwealth]] ([[User talk:Houseofwealth|talk]]) 01:11, 16 January 2020 (UTC)

==Error in limit definition==

Greetings maintainers of this page. In the section "Definition in terms of limits of functions" you give the standard definition as 

<math>\displaystyle \lim_{x \to c} f(x) = f(c)</math>. This is correct. Two paras later you say:

"The formal definition of a limit implies that every function is continuous at every isolated point of its domain."

This does not follow. In fact in the Wikipedia article on limits of functions at https://en.wikipedia.org/wiki/Limit_of_a_function, in the section "More general subsets," it is explicitly stated that limits can only be defined at limit points. That's because if you try to define limits at isolated points, the limits are not unique. That latter point is not made explicitly in the article, but it's the reason that limits are only defined at limit points.

Therefore, continuity at isolated  points does NOT follow from the limit definition; but rather only from the epsilon-delta definition.

[[Special:Contributions/99.171.179.102|99.171.179.102]] ([[User talk:99.171.179.102|talk]]) 21:14, 20 February 2021 (UTC)

==Please remove the "highschool definition"==
"The most common and restrictive definition is that a function is continuous if it is continuous at all real numbers. In this case, the previous two examples are not continuous ..."

and:

"... but is not continuous over the domain <math>\R</math> because it is undefined at x = 0"

What is this noobish nonsense? <math>f(x)=1/x</math> is a continuous function PERIOD - A function cannot be (dis-)continuous "over the domain <math>\R</math>" if it is not defined at every <math>x\in\R</math>. If something is defined in a wrong way in highschool (because it may be too complicated otherwise, I guess) this should not result in a Wikipedia article containing said wrong definition (as viable).

An example that is perfectly analogous to the above case:
"The function <math>f(x)=x^2</math>, <math>x\in\R</math> is not a continuous function over the domain <math>\Complex</math> because it is undefined for <math>x\in\Complex/\R</math>."

I really hope it is clear that this is nonsense.
--[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 12:56, 10 December 2021 (UTC)
: I have clarified the definition(s) of (global) continuity. In particular, it depends on the context whether <math>f(x)=1/x</math> is considered as a continuous function. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:12, 10 December 2021 (UTC)
::I'll cite somebody from above on a similar example:"If someone imagines tan(x) to not be continuous, they have not quite got it".
::A function can only be continuous or discontinuous at points of its domain.  "(1/x) ... is discontinuous at <math>{\displaystyle x=0}</math>" is wrong just the same way it is wrong that <math>f(x)=x^2</math>, <math>x\in\R</math> is (not) continuous at e.g. <math>2+i.</math> 
::I know there are many who really want to cling to the idea of continuity of functions as whether they can be drawn without resetting the pen. But this is not in line with modern day mathematics. --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 21:11, 12 December 2021 (UTC)
:::In the context of [[calculus]], "function" means often "partial function", and the domain of a function is often unknown, or, at least, difficult to compute. For example, are you able to describe without a computer the domain of <math>f(x)=1/(x^6-4x^5+3x^4-7x^3+x^2-1)</math>? Another example is the complex function <math>f(z)=1/\Gamma(z),</math> where <math>\Gamma</math> denotes the [[gamma function]]; if you are able to describe its domain, congratulation, you will win the [[Millenium Prize Problems|Millenium Prize]] for having solved the [[Riemann hypothesis]]. An opposite example is the function <math display=inline>f(x)=\frac{x^2-2x+1}{x^2-1},</math> which is not defined for <math>x=1,</math> but becomes continuous if one defines <math>f(1)=0.</math> 
:::All these examples (but the last) are continuous in their domain, but cannot be prolongated in a function that is continuous in a larger domain. For this reason, modern mathematicians say that these functions are continuous except at the zeros of their denominators, and these zeros are called ''discontinuities'' or ''singularities'', or, in these particular examples [[zeros and poles|poles]]. 
::: In summary, this is you who are not in line with modern mathematics (modern mathematics are mathematics of living mathematicians, not necessarily mathematics taught in elementary calsses). [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 22:37, 12 December 2021 (UTC)
::::I doubt that most modern mathematicians would agree with the statement "<math>f(x)=1/x</math> ... is discontinuous at <math>{\displaystyle x=0}</math>" simply because it is plain wrong. It does not agree with the definitions of continuity given in this very article (- all definitions use the value of the function at the respective point, i.e. it has to be in the domain). I also doubt they treat the terms "singularity" and "discontinuity" as equivalent, in general. And if in certain areas of mathematics where mostly functions, the domain of which is difficult to evaluate, are considered and because of that, the term "continuous function" is used differently, then that should warrant a subheading within this article where this is specifically discussed. Also, could you please point me to review articles or similar sources where the term is used in the way you described it? --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 18:23, 21 December 2021 (UTC)
:::::Ok, apparently there have been some changes to the article in favor of my argument, thank you. There is still:
:::::1. "<math>f(x)=1/x</math> ... is discontinuous at <math>{\displaystyle x=0}</math>" - see above (or below)
:::::2. On functions with singularities: "When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested with their behavior '''near the exceptional points''', one says that they are discontinuous."
:::::- '''Near''' poles these functions are continuous and, again, '''at''' the poles they are not defined, hence, neither continuous nor discontinuous.
:::::And the first sentence (of 2.), I think, just adds confusion to the topic. Because it is spot on, let me repeat the quote from above:"If someone imagines tan(x) to not be continuous, they have not quite got it"
:::::Continuity of functions is about wheter they "respect" the limes, not about drawing the graph of a function in one uninterrupted line. This needs to be a fundamental message of the article.
:::::--[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 12:25, 8 January 2022 (UTC)
:::::Since there has been no answer, can we/I please remove/change the parts of the article talking about discontinuities outside the domain of functions? This includes the graph of f(x)=1/x including the wrong statement "... but is discontinuous at <math>x=0</math>" - Or we could keep the graph but with a paragraph talking about the very common mistake of this and other functions with singularities being discontinuous there. E.g. the French article on this subject has such a paragraph. And, btw, neither the German, the French nor the Spanish article claim these functions have discontinuities. It is a widespread misconception and this article should help correcting it, not the opposite. --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 15:14, 20 May 2022 (UTC)
::::::One can find, in many reliable sources sentences such that {{tqq|The function <math>x\mapsto \frac 1x</math> is discontinuous at <math>x=0.</math>}} It is not the role of Wikipedia to decide whether this formulation is wrong or not, but, as such a formulation is very common, Wikipedia ''must'' report it. I agree that the section deserves to be improved or clarified, and, as a first step in this direction, I have added a link to [[Partial function]]. As your peremptory assertions are unsourced, they are considered here as your own [[WP:OR|original research]]. So, please do not base edits on them, this is against one of the main Wikipedia policies (see [[WP:NOR]]). [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:55, 20 May 2022 (UTC)
:::::::What I said is not original research. I already mentioned the French article and found in Arens, Hettlich - Mathematik (2018), page 219, for exactly the same example function (translated from German):"One can only speak of continuity in places where a function is defined". And I asked before if you could point me to sources stating the opposite but there was no answer. So in which mathematics textbooks (or other reliable sources) is it written that "The function <math>x\mapsto \frac 1x</math> is discontinuous at <math>x=0</math>" (or similar, e.g. different function with singularity, with the same meaning)? --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 16:27, 24 May 2022 (UTC)
:::::::: The article [[Zeros and poles]] starts with "In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly", and it is later said that 0 is a pole of <math>1/x.</math> You can find similar formulations in every textbooks of complex analysis. So the fact that in some contexts 0 is a singularity of <math>1/x</math> is a sourced fact. Your peremptory assertion that this is wrong is thus your own opinion that is not a consensus of the mathematical community. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 18:17, 24 May 2022 (UTC)
:::::::::Yes, but this was never in question. The question was whether a pole/singularity is a discontinuity. I say, it is not, you say it is. You said in the post before your last one, that:"One can find, in many reliable sources sentences such that {{tqq|The function <math>x\mapsto \frac 1x</math> is discontinuous at <math>x=0.</math>}}" - Where are they? I offered 2 sources that this is wrong, regarding even exactly the same example. --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 08:35, 27 May 2022 (UTC)
:::::::::This keeps getting better. Now you added complex analysis in the section "Real functions"-->"Definition"- again, without source (and with typos/missing articles). And the Cauchy Prinicipal Value part, you added as well, does not at all show that functions with singularities are discontinuous there - this concept allows for integration over singularities - why is this in the introductory section of the article "Continuous Function"? The part on partial functions also does not argue for functions with poles being discontinuous there, is phrased vaguely and also has a typo: "In other contexts, mainly when one is interested WITH their behavior near the exceptional points, one says that they are discontinuous."--> (I said this above already, btw):"'''near the exceptional points'''" they are continuous anyways (!), it is nowhere to find in the article "Partial Functions" that they are discontinuous '''at''' the poles and, so far, there are no sources backing it. - And again, where are your "many reliable sources sentences such that "The function <math>x\mapsto \frac 1x</math> is discontinuous at <math>x=0.</math>"? Even if you come up with them (which I doubt for this example), I already gave two sources saying that this is wrong and I'm quite sure I can bring more. --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 09:15, 2 June 2022 (UTC)
::::::::::Again, there is no answer and no sources for your claim, so, so far, this is original research. I have provided two sources backing my position. So I repeat what I wrote above:"Can we/I please remove/change the parts of the article talking about discontinuities outside the domain of functions? This includes the graph of f(x)=1/x including the wrong statement "... but is discontinuous at <math>x=0</math>" - Or we could keep the graph but with a paragraph talking about the very common mistake of this and other functions with singularities being discontinuous there. E.g. the French article on this subject has such a paragraph. And, btw, neither the German, the French nor the Spanish article claim these functions have discontinuities. It is a widespread misconception and this article should help correcting it, not the opposite".
::::::::::In addition, the complex analysis-part you added in the section "Real functions" should be removed for an obvious reason. --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 12:23, 17 July 2022 (UTC)

{{unindent}}Umm, how to put it politely: that's not correct. <math>f(x)=1/x</math> is a standard textbook example of a function that is not continuous at x=0. It appears in pretty much all standard introductory pre-calculus textbooks used in high-schools throughout the land. Its a standard example, precisely because, to the young student, it looks like a nice smooth function, and so whenever some kid starts getting glib about continuity, you ask them to do a delta-epsilon proof at x=0, and they can't. Try it yourself. The entire ''raison d'etre'' of theorems and proofs is that, by being forced to write a detailed proof, you avoid glib but incorrect claims about this-and-such being true. You have to actually prove it. Try it. You'll fail. There is no proof of continuity at x=0.  [[Special:Contributions/67.198.37.16|67.198.37.16]] ([[User talk:67.198.37.16|talk]]) 06:25, 29 December 2023 (UTC)

:You are mistaken. For the epsilon-delta criterion (or the test for continuity via limits) at a position, the function has to be defined there. This is not the case for <math>f(x)=1/x</math> at x=0 (and poles of functions in general). I went over this numerous times above - did you read that at all? Note that D.Lazard, in an attempt to justify his erratic notion of continuity, produced citation 7 (currently): Strang, Gilbert (1991). Calculus, which says:
:"Objection: The definition makes <math>f(x)=1/x</math> a continuous function! It is not defined at x=0, so its continuity can't fail. The logic requires us to accept this, but we don't have to like it."
:Isn't that interesting? He, himself, gave a source that directly contradicted his view. Further, it's not about whether "we like it" or not, as it says in the source. It's about what is correct. And correct is that a function is continuous if it is continuous at every position of its domain - which is the case for <math>f(x)=1/x</math>. And a discontinuous function has at least one discontinuity - but it can only be discontinuous where it is defined. It's part of the definitions of both the limits- and the delta-epsilon-criterion. If some school- and undergrad books ignore these definitions and arrive at the conclusion that <math>f(x)=1/x</math> is a discontinuous function, then they are wrong in this regard, which (being wrong) isn't that special for such books.
:Btw I found that D.Lazard also wrote inaccurate content in the Wikipedia article on the square root, where he implies that the square root of e.g. 4 is {-2, 2}. He also contributed to the article "Multa-valued function", which doesn't exist in Wikipedias of other languages - simply because it doesn't exist at all. Apparently not enough mathmaticians/scientists who care for accurate content care enough for it to contribute to Wikipedia pages. This is very problematic because Wikpedia has such a wide reach. --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 00:21, 12 April 2024 (UTC)
::Please just use direct language instead of sarcasm or rhetorical flourishes, and try to stay polite and assume good faith. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 01:27, 12 April 2024 (UTC)
:::Could you please point me to where was I less polite and more sarcastic than "Umm, how to put it politely:"?
:::And regarding assuming good faith: In the original dispute from 2021/2022 with D.Lazard, he just stopped replying to my arguments that he could not counter and ignored the source I gave to back my standpoint - and his own source (Strang, Gilbert (1991)) contradicts his own standpoint. Yet, he didn't revert his changes (e.g. where now there is complex number content under the heading "Real Functions") to the article. --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 11:03, 12 April 2024 (UTC)
::::I stopped replying, because I had replied before, and I do not like to waste my time to repeat the same thing again and again. About self-reverting my edits, I would do it if there were a consensus for that. In this case, you alone is certainly not sufficient for a consensus. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 11:41, 12 April 2024 (UTC)
:::::Yes, you replied, but you didn't address any of my objections. So it had the same effect as no reply. --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 11:03, 15 April 2024 (UTC)
::::Parts such as "did you read that at all?", "an attempt to justify his erratic notion", "Isn't that interesting?", and "Apparently not enough mathmaticians/scientists who care for accurate content care enough for it to contribute to Wikipedia pages." seem entirely gratuitous to me. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 12:26, 12 April 2024 (UTC)
:::::Dear @[[User:Jacobolus|Jacobolus]],
:::::I do not believe that his comments are very sarcastic. @[[User:Felix Tritschler|Felix Tritschler]] (who, it seems, has left) was the only one for some time who consistently cited proper sources. He took the time to go through @[[User:D.Lazard|D.Lazard]]'s source and found the citation (which actually claims the opposite of what Lazard suggests).
:::::Therefore, I'd like to ask again if anyone has a calculus textbook that states that \( \frac{1}{x} \) is discontinuous at 0. To summarize the discussion:
:::::a) Yes, it has a pole. The three kinds of singularities are discussed in every complex analysis book.  
:::::b) Yes, the pole is not a removable singularity.  
:::::c) The statement, "When they are continuous on their domain, one says, in some contexts, that they are continuous," is incorrect. Every calculus textbook says otherwise, and no contrary source has been provided. Most textbooks explicitly discuss \( \frac{1}{x} \).  
:::::d) Therefore, the caption in the first figure is incorrect. The cited source actually says something different; one just needs to continue reading after the section on "poles" to the subsection on "Continuous function."  
:::::e) In the past, \( \frac{1}{x} \) had its own subsection in Wikipedia, explaining that many people have a misconception about it. Unfortunately, the content of this article has since changed to suggest the opposite.
:::::If no one has supporting evidence for point c), can we please edit it? As it currently stands, it is incorrect, and there is no citation to back up the claim that calculus textbooks state \( \frac{1}{x} \) is discontinuous. I could use parts of this discussion in my "Didactics of Calculus" lecture for teacher education, as it provides prime examples of misunderstandings. And yes, many school textbooks are incorrect about continuity (this was, for example, a topic in Daniel Frohn’s talk at GDM 2024, though the paper is not yet available online).
:::::I, therefore, suggest a compromise: we begin with the correct definition from calculus and also mention the discussions that regularly arise when extending the function on \( \mathbb{R} \). [[User:Math Michi|Math Michi]] ([[User talk:Math Michi|talk]]) 09:56, 4 September 2024 (UTC)
::::::There are many sources which define real-valued functions to be discontinuous at isolated points not included in their domain. For example:
::::::: "All elementary functions are continuous in their domains of existence. The points of discontinuity do not belong to the domain of existence... {{pb}} "Rational functions {{tmath|\tfrac{P(x)}{Q(x)} }} ({{tmath|P(x)}} and {{tmath|Q(x)}} are polynomials) are continuous everywhere except those values of {{tmath|x}} for which {{tmath|1= Q(x)=0 }} but {{tmath|P(x)\neq 0}}; if {{tmath|1= x = a}} is a root of the denominator, but {{tmath|P(a) \neq 0}}, then the function has an infinite discontinuity for {{tmath|1= x = a}}. If {{tmath|a}} is a root of both the numerator and denominator, then the function has an infinite discontinuity only in the case when the multiplicity of the root in the denominator is greater than that in the numerator; otherwise the discontinuity is removable."
::::::From {{doi|10.1007/978-1-4684-6288-3_9}}. {{pb}} Or since you wanted a published introductory calculus book, here's one which adopts this terminology, {{doi|10.1007/978-1-4615-6594-9}}. {{pb}} {{tq|i=yes|"incorrect"}} – this is a purely semantic quibble; what you are calling "incorrect" is just a different linguistic convention than your preference, without much substantive mathematical difference. Wikipedia's job here is to describe common usage, not to prescribe one Wikipedian's preferred definitions. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 15:12, 4 September 2024 (UTC)
:::::::Dear, @[[User:Jacobolus|Jacobolus]]
:::::::Both books are a) quite old and b) not available online. I believe both points are important to consider in this discussion, as we also don't use schoolbooks from the 1970s anymore in the discussion. Additionally, for a constructive discussion, easily accessible sources are preferable.
:::::::However, I was able to get hold of Knight & Adams (1975), which is clearly not a Real Analysis book. Covering the concepts of limits and continuity in just one lecture, as suggested, is less time than some school systems dedicate to these topics. The entire book, upon reviewing it, is written in a pre-formal style (Blum & Kirsch, 1991). Concepts like the domain are presented in an explanatory manner, not in a formal one. However, the arguments here focus on a function and its domain, and so on.
:::::::The book has been cited 0 times. With all due respect to Knight & Adams, this book does not serve as relevant support for the claim. Additionally, I couldn't find any other significant contributions from these authors to mathematics.
:::::::Of course, you may present this book's perspective on Wikipedia, but it should be clearly noted as a minority view. In contrast, the referenced and cited Strang work is 20 years newer, has 275 citations, and the author is a recognized scholar and scientists. He explicitly contradicts the interpretation of 1/x.
:::::::I was unable to find your other source, but I did come across a similar book by the same authors: Bronštejn, I. N., & Semendjaev, K. A. (2013), Handbook of Mathematics, which appears to be essentially the same. The definition of continuity in that book differs from the one you cited.
:::::::"Definition of Continuity
:::::::A function \( y = f(x) \) is called continuous at the point \( x = a \) if:
:::::::1. \( f(x) \) is defined at \( a \)
:::::::2. \( \lim_<nowiki>{{x \to a}}</nowiki> f(x) \) exists and is equal to \( f(a) \)"
:::::::They continue:
:::::::"A function has a point of discontinuity at \( x = a \), which is an interior point or an endpoint of its domain, if the function is not defined there, or \( f(a) \) is not equal to the limit \( \lim_<nowiki>{{x \to a}}</nowiki> f(x) \), or the limit does not exist." 
:::::::This is a (I'd say) possible way, to discuss functions  on \( \mathbb{R} \). It's followed by a little bit later "the points of discontinuity do not belong to their domain" in 2.1.5.4 which you are citing. Therefore I want to point out multiple things.
:::::::a) You wrote in discussions here that \( \frac{1}{x} \) is discontinuous because the right- and left-handed limits are different. You have now cited a book that does not back up that claim.
:::::::b) I would like to add that the book uses "discontinuities" often as a Noun to describe the local behavior at \( x = a \) and not as a global property of \( f \), continuous being an adjective. The definitions are mainly used to create a framework where "different behaviors" at "interesting points" can be investigated. While I understand the didactical idea behind this, I do not agree that \( f(x) = 1 \) on \( A = (0,1] \) has a discontinuity at 0. In this framework, it does (because the framework implicitly "desires" to investigate the function on \( \mathbb{R} \) and not just \( A \)). Correct me if I am wrong.
:::::::c) It is not common, at least in Austria, Germany, or in some Japanese textbooks I am familiar with, to discuss functions like \( \frac{1}{x} \) on the domain \( \mathbb{R} \) in school (however one introduces that concept). I am also confident that if I were to open a textbook from the US or UK, \( \frac{1}{x} \) would be defined on \( \mathbb{R} \setminus \{0\} \).
:::::::d) Following this, the desire to introduce partial functions does not improve readability for students and is also inconsistent with most Real Analysis literature.
:::::::So, why should we do this? Just because someone wants to call a pole of the second type a discontinuity? [[User:Math Michi|Math Michi]] ([[User talk:Math Michi|talk]]) 13:59, 5 September 2024 (UTC)
::::::::You can find hundreds of examples of this kind of description, which is in widespread use. You are welcome to do an exhaustive survey if you like. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 14:17, 5 September 2024 (UTC)
:::::::::This short answer shows quite clearly why Felix, for example, has stopped writing long explanations. No matter how many arguments are put forward, if they don't fit, they are not addressed.
:::::::::Instead, you react like this when the arguments you put forward here are repeatedly countered with some of the works you yourself have mentioned. I took the time to find and read them while in your answer, you don't even make it clear what exactly you mean. Many textbooks? Many real analysis books? [[User:Math Michi|Math Michi]] ([[User talk:Math Michi|talk]]) 14:46, 5 September 2024 (UTC)
::::::::::You keep moving the goalposts. You asked for a calculus textbook so I linked you to a calculus textbook. But now apparently that's not good enough because it's too old and now you want a real analysis textbook.
::::::::::You thought the previously mentioned sources were too old, so how about this example from 2008, {{doi|10.1007/978-0-387-48807-3_8}}: "The linear function {{tmath|bx + c}} is defined for all values of the argument {{tmath|x}} and (unless {{tmath|1=b =0}}) itself assumes all values. The same is true of the reciprocal linear function which, however, displays an infinite discontinuity at {{tmath|1= x = -c/b}}" (p. 58) "When the discriminant is zero, the reciprocal quadratic function has the unusual property of encountering a infinite discontinuity of the {{tmath|{+\infty}{{!}}{+\infty} }} type at {{tmath|1=\textstyle x = -\sqrt{c / a} }}." (p. 132).
:::::::::: This second edition book has 692 citations in Google Scholar, alongside the 1200 citations of the first edition, so not being cited enough hopefully won't be enough reason for you to reject this one. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 15:19, 5 September 2024 (UTC)
::::::::::Here's an example of a book entitled ''Introduction to Real Analysis'', 2009, which uses this kind of language [https://books.google.com/books?id=ab-2vpx0FyYC&pg=PA8 on page 8]: ""Subclassification of an essential discontinuity is based on whether the one-sided limits are unbouned, called an ''infinite discontinuity'', or bounded, called an ''oscillatory discontinuity''." –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 15:31, 5 September 2024 (UTC)
:::::::::::Thank you for the many sources provided. We seem to be going around in circles here. I will repeat what I have written elsewhere: many of the examples cited refer to the point of discontinuity as an object of the sentence, giving it the name "discontinuity," but do not attribute the property of being discontinuous to the function ''f'' itself, which is an adjective.
:::::::::::You are correct that it is possible to criticize my rejection of the source, but I believe that my arguments for why I do not consider it relevant are understandable.
:::::::::::However, your sources contradict you in other places. The example you mentioned, ''Introduction to Real Analysis'' (2009), contains Example 1.4.2 and Theorem 1.5. The latter states: "Polynomials are continuous everywhere. Rational functions are continuous wherever defined; the discontinuities of rational functions are either removable or infinite."
:::::::::::Therefore, 1/x is a continuous function (adjective) and has a discontinuity at x=0. For the latter, many other authors choose different terminology.
:::::::::::All of the examples cited fall into this category. They always deal with the point x=0, but not with the function ''f''. However, we are discussing continuity of a function, not "points of discontinuity." If you are unwilling to see this distinction, we will not reach an agreement.
:::::::::::Indeed, this is somewhat like the situation where "the set is both open and closed," when I say that a function is continuous on its domain but has a point of discontinuity.
:::::::::::''An Atlas of Functions'' does not seem to define continuity. Once again, it discusses the point and not the property of the function.
:::::::::::''Resources For The Study of Real Analysis'', in the example you mentioned, defines the points as "jump discontinuities," again not referring to the property of the function. I have spent some time searching, but I cannot find pages 36-37, where the definition of a continuous function is located. However, this is the focus of the Wikipedia article.
:::::::::::None of your sources has claimed, as we do in the article, that "1/x is discontinuous at x=0"; rather, they give the point x=0 a name. [[User:Math Michi|Math Michi]] ([[User talk:Math Michi|talk]]) 14:02, 14 September 2024 (UTC)
::::::::::Here's ''Resources for the Study of Real Analysis'', 2004, published by the MAA, [https://books.google.com/books?id=IBrzs1DqzegC&pg=PA38 p. 38]:
::::::::::"The function {{tmath|1= g(x) = \bigl[\tfrac1x\bigr]}} has all its discontinuities in the bounded interval {{tmath|[-1, 1]}} and has an infinite discontinuity there as well." (Here the brackets are used for the [[floor function]].) –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 15:38, 5 September 2024 (UTC)
::More productively, if you want to find sources, one keyword to try is "infinite discontinuity", which will turn up many textbooks, papers, university course notes, etc. (though sometimes this means something else). –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 13:00, 12 April 2024 (UTC)
:::Dear @[[User:Jacobolus|Jacobolus]], what specific claim are you referring to? I searched on Google Scholar and Google, and everything I found seemed to confuse the concept of the type of a pole with the concept of continuity. [[User:Math Michi|Math Michi]] ([[User talk:Math Michi|talk]]) 10:19, 4 September 2024 (UTC)

In reply to @[[User:D.Lazard|D.Lazard]]'s  [[#c-D.Lazard-2022-05-20T16:55:00.000Z-Felix_Tritschler-2022-05-20T15:14:00.000Z|comment, 16:55, 20 May 2022]]:
: @[[User:Felix Tritschler|Felix Tritschler]] pointed out, that your source Strang, Gilbert (1991). Calculus. SIAM. p. 702. ISBN 0961408820 states the opposite to your claim.
:Can you please name another backup to your claim e.g. state another 'reliable source'
:Thank you in advance [[User:Math Michi|Math Michi]] ([[User talk:Math Michi|talk]]) 10:30, 4 September 2024 (UTC)

::This is an answer to a misplaced post. Apparently, for you, the two following sentences are not equivalent: "{{mvar|f}} has a discontinuity at {{math|1=''x'' = 0}}" and "{{mvar|f}} is discontinuous at {{math|1=''x'' = 0}}". Can you provide a source that asserts that the sentences are not equivalent? Can you explain in a few lines why you think that the sentences must have different meanings (if your explanation requires more than, say, five lines, you can be sure that almost nobody will read it carefully, and in any case, the explanation would be too complicate for being included in the article)? [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:49, 14 September 2024 (UTC)
:::Thank you for your answer.
:::My claim is, that the sentence "''f'' has a discontinuity at ''x= 0''" is about the singularity or the pole
:::And does not mean (by the cited authors) " ''f'' is discontinuous".
:::My data&backup for this claim are primarly the books you and @[[User:Jacobolus|Jacobolus]] brought up:
:::''Introduction to Real Analysis'' (2009) says "Rational functions are continuous wherever defined"
:::''Strang, Gilbert (1991). Calculus. SIAM. p. 702. ISBN 0961408820'' says:
:::"The definition makes f(x)=1/x a continuous function! It is not defined at x=0, so its continuity can't fail." 
:::''Bronštejn, I. N., & Semendjaev, K. A. (2013), Handbook of Mathematics''  says:
:::Definition of Continuity
:::A function '' y = f(x) '' is called continuous at the point '' x = a '' if:
:::1. '' f(x) '' is defined at '' a ''
:::2. ''\lim_{x \to a} f(x)'' exists and is equal to ''f(a) ''
:::followed by "the points of discontinuity do not belong to their domain"
:::Therefore yes @[[User:D.Lazard|D.Lazard]] I see this as backup that 
:::a) "''f'' has a discontinuity at ''x'' = 0" and "''f'' is discontinuous"
:::is not the same
:::b) I repeat what I wrote here multiple times, which (relevant) text-book has stated, that 1/x or a comparable function is discontinuos. Yes, multiple books have written, that it has a discontinuity at a pint ''x''. Which is about a local property-description of the pole and not a global property description of ''f'' [[User:Math Michi|Math Michi]] ([[User talk:Math Michi|talk]]) 19:58, 14 September 2024 (UTC)
::::Certainly, {{tqq|1=f has a discontinuity at x = 0" and "f is discontinuous" is not the same}}. But {{tqq|1=f has a discontinuity at x = 0" and "f is discontinuous at x = 0" is the same}}. So, what precedes is not an answer to my question. This shows only that the phrase "''f'' is discontinuous" should not be confused with "f is discontinuous at a given point". [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 20:57, 14 September 2024 (UTC)

== Suggestion to change the article based on the discussion 'Please remove the "highschool definition" ' ==
Since there has been no reaction, specifically no sources backing that functions are discontinuous at singularities etc. (see above), I am suggesting here anew to change the article accordingly, that is, to clarify that functions are neither continuous nor discontinuous at singularities or at any other points outside their domain and, hence, that functions like <math>x\mapsto \frac 1x</math> or the tangent function are continuous functions (A function that is continuous at every point in its domain is a continuous function). In the above discussion I have given a source from a mathematics textbook (and the French article on continuous functions). Btw, these issues with the article have been addressed before, in the now archived discussion "Continuity and the domain of the function" where it is correctly stated that "elementary functions are continuous" but either the article was never changed in this regard or these changes were reverted. (Plus, there have been very questionable recent additions and changes to the article like content on complex analysis in the section "Real functions".)  I would encourage other users, especially the contributors of the now archived discussion, to get involved. Thank you in advance.
--[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 11:50, 6 August 2022 (UTC)

:You are proposing something that is contradicted by pretty much every textbook on pre-calculus used throughout the land. The reason this is a "high-school definition" is because it is used in high-schools. It is what students see, it is what is taught by convention. It appears on exams and in homework problems. [[Special:Contributions/67.198.37.16|67.198.37.16]] ([[User talk:67.198.37.16|talk]]) 06:31, 29 December 2023 (UTC)
::Please have a look at my reply to your other comment (IP address 67.198.37.16 from 29.Dec.2023). --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 11:05, 12 April 2024 (UTC)
::Likewise, the current version using terms like "partial function" and "topological closure" is far from being suitable for a high-school level definition. However, you are correct that most school-textbooks are wrong on this matter. 
::It is therefore the responsibility of Wikipedia to:
::a) provide a correct definition,  
::b) point out that many textbooks contain inaccuracies,  
::c) ideally, cite a didactics article that explains the origin of this misconception. [[User:Math Michi|Math Michi]] ([[User talk:Math Michi|talk]]) 10:08, 4 September 2024 (UTC)

== Definition in terms of neighborhoods has issues ==

The definition begins with: "A neighborhood of a point c is a set that contains, at least, all points within some fixed distance of c." This is true for a metric space but there are topological spaces that are not metrizable.

The definition continues: "Intuitively, a function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point  as the width of the neighborhood around c shrinks to zero." Again, the "width" is only well defined in a metric space.

It ends with "This definition only requires that the domain and the codomain are topological spaces and is thus the most general definition." This may be the case if we replace "neighborhood" by "open set" but given that neighborhoods are introduced as "points within some fixed distance", it is plainly wrong. [[Special:Contributions/120.51.141.38|120.51.141.38]] ([[User talk:120.51.141.38|talk]]) 05:51, 17 April 2023 (UTC)

:As this section is a subsection of {{alink|Real functions}}, there is no problem, except in the last paragraph, which I have fixed by mentioning that this is a generalization of the concept of neighborhood that is used there. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 09:01, 17 April 2023 (UTC)

== 1/x is discontinuous at 0? ==

I know that some calculus textbooks does say that <math>x\mapsto 1/x</math> is discontinuous at 0 since it is not defined there. But this is a kind of old way to describe the matter and the map <math>x\mapsto 1/x</math> becomes continuous (even [[meromorphic]]) after defining <math>1/0=\infty</math>. If we are to keep this example in the article I think it's best to replace the function by a map
:<math>f(x)=
\begin{cases}
1/x&x\ne0\\
0&x=0
\end{cases}
</math>
which is also defined at 0. [[User:慈居|慈居]] ([[User talk:慈居|talk]]) 08:42, 28 October 2023 (UTC)

:And by the way, I'm here because this confusing statement has arrived at Korean Wikipedia by translation. [[User:慈居|慈居]] ([[User talk:慈居|talk]]) 08:47, 28 October 2023 (UTC)
::The article says {{tqq|Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some isolated points. Examples are the functions <math>x\mapsto 1/x</math> and <math>x\mapsto \tan x.</math> When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are discontinuous. [...] For example, the functions <math>x\mapsto 1/x</math> and <math>x\mapsto \sin \frac 1x </math> are discontinuous at 0, and remain discontinuous whichever value is chosen for defining them at 0.}}. I think that this answers your concern. Also, when not otherwise specified, <math>x\mapsto 1/x</math> is a (partial) function from the reals to the reals or from the complexes to the complexes. Its extension as a function on the [[projectively extended real line]] or the [[Riemann sphere]] is continuous everywhere, but it is another function. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 12:11, 28 October 2023 (UTC)
:::Oh, that definition for partial functions is even a special case of continuous functions in the normal sense, by adding a new isolated point to the codomain. This is indeed enough to make the statement make sense. But as for my home wiki, I'd rather not introduce this definition and only discuss continuity at points in the domain, cause that seems enough. [[User:慈居|慈居]] ([[User talk:慈居|talk]]) 13:48, 28 October 2023 (UTC)
:::First, I doubt that the consensus is that, when not otherwise specified, <math>x\mapsto 1/x</math> is assumed to be a partial function. But even if, that does not produce a discontinuity at x=0: Neither via the epsilon delta criterion, nor via limits is it possible to determine a discontinuity at x=0. As a partial function, x=0 still does not belong to the domain of this partial function and the criteria for discontinuity can only be applied at points of the domain of a (partial) function.
:::Please have a look at the source "Strang, Gilbert (1991). Calculus. SIAM. p. 702. ISBN 0961408820", where it says:
:::"The definition makes f(x)=1/x a continuous function! It is not defined at x=0, so its continuity can't fail." --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 11:33, 12 April 2024 (UTC)
::::The meaning of the expression {{tmath| x \mapsto 1/x}}, without further explicit detail, depends on context and is somewhat fuzzy and not always consistent. In complex analysis, dealing with arbitrary rational functions, it is most convenient to be working with the projectively extended complex numbers, {{tmath|1= \hat\C = \C \cup \{ \infty \} }}, so naming this function {{tmath|f}} we have {{tmath|1= f(0) = \infty}} and is continuous there. In applications working with real-variable calculus, it's most common in practice to consider partial functions from {{tmath| \R \to \R}}, so that {{tmath|f(0)}} is considered undefined and is a discontinuity because {{tmath|1=\textstyle \lim_{x \to 0^+} f(x) \neq \lim_{x \to 0^-} f(x)}}. In computer floating point arithmetic the number system includes signed zero (representing either 0 or possibly underflow) and signed infinity (representing either infinite quantities or overflow), so {{tmath|1=f(0) = \infty}} and {{tmath|1=f(-0) = -\infty}}, both well defined but clearly a discontinuity. Some authors, especially of introductory textbooks, work only with functions per se, with the domain defined to exclude values where the function is undefined. For these authors, {{tmath|f(0)}} is undefined and {{tmath|1=\textstyle \lim_{x \to 0^+} f(x) = +\infty}}, {{tmath|1=\textstyle \lim_{x \to 0^-} f(x) = -\infty}}, but the function's domain is said to be {{tmath|(-\infty,0)\cup (0,\infty)}} (as is the function's image) and the function is said to be continuous for points in this (itself non-continuous) domain. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 17:19, 12 April 2024 (UTC)
:::::As a real valued function of a real variable, isn't <math>1/x</math> most commonly a function <math>\mathbb R\setminus\{0\}\to\mathbb R</math>? [[User:慈居|慈居]] ([[User talk:慈居|talk]]) 17:41, 12 April 2024 (UTC)
::::::Perhaps @[[User:Jacobolus|Jacobolus]] wanted to define an additional bijective (and, as we know, continuous) function. I agree with you, typically the function is <math>f: \mathbb{R} \setminus \{0\} \to \mathbb{R}</math>. [[User:Math Michi|Math Michi]] ([[User talk:Math Michi|talk]]) 12:53, 4 September 2024 (UTC)
:::::My impression is that defining continuity at an undefined point is rather rare. And the only reference that I know that does so, defines ''every'' function <math>\mathbb R\setminus\{0\}\to\mathbb R</math> to be discontinuous at 0, because it is not defined at 0, which has nothing to do with analysis at all. [[User:慈居|慈居]] ([[User talk:慈居|talk]]) 17:58, 12 April 2024 (UTC)
::::::Continuity has to do with whether the limit from the left is the same as the limit from the right and whether these equal the value of the function. In the case where the function is undefined at a particular point, it is indeed usually taken to be discontinuous there, but sometimes there is a [[removable singularity]] which can be trivially fixed by just defining the function at that point to be equal to its limit (e.g. starting from {{tmath|x \mapsto \sin x / x}} we can explicitly define the value at {{tmath| 0 \vphantom) }} to obtain the continuous [[sinc function]] even though ordinarily division by zero is undefined), while other times there is some kind of "jump" where the limits from right and left do not agree, or a place where the function "blows up" and tends to infinity (from one or both sides, with the same or differing signs from above and below), or even an [[essential singularity]] where the behavior is more pathological. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 20:39, 12 April 2024 (UTC)
:::::::Such classification is made because we often consider a function from and to a subset of <math>\mathbb C</math>, but we also wish to see their behavior as a map from and to a subset of <math>\widehat\mathbb C</math>. But actually there is only one concept of "[[holomorphic function|holomorphy]]" for a fixed domain and codomain. Holomorphy is defined for points in (the interior of) the domain and I think continuity should be too. [[User:慈居|慈居]] ([[User talk:慈居|talk]]) 05:53, 13 April 2024 (UTC)
::::::::In the original source which introduced the name "holomorphic function", Briot & Bouquet's ''Théorie des fonctions elliptiques'' (2nd edition, 1875), it is [https://archive.org/details/thoriedesfonct00briouoft/page/14/ taken for granted] that we can take the reciprocal of a function which has a pole at a point and get a zero there, or vice versa. That is, the pole of the function is at least implicitly considered to be part of the domain (or something close enough to the domain that it can counts as part of the domain of the reciprocal function). –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 06:09, 13 April 2024 (UTC)
:::::::::I see it but I didn't get what your point is. Another way to say this would be that a function holomorphic at points except some isolated poles has an [[analytic continuation]] to a domain that contains those poles, but valued in <math>\widehat\mathbb C</math>. The reciprocal function also has a continuation to a domain containing the infinity, and the two maps can be composed freely. [[User:慈居|慈居]] ([[User talk:慈居|talk]]) 06:23, 13 April 2024 (UTC)
:::::::::++ I'm fine with assuming that the domain already contains the poles, or not assuming it, which varies among different authors. I'm only concerned with defining continuity only at points in the domain. [[User:慈居|慈居]] ([[User talk:慈居|talk]]) 06:47, 13 April 2024 (UTC)
::::::::::The current article text seems quite clear and explicit, discussing both points of view. Are you just wondering about finding sources for each version? For example, a 2021 version of [https://abel.math.harvard.edu/~knill//teaching/math1a2021/handouts/lecture04.pdf introductory calculus notes] from Harvard have "The function {{tmath|1=f(x) = 1/x}} is continuous except at {{tmath|1= x = 0}}. It is a prototype with a pole discontinuity at {{tmath|1= x = 0}}. One can draw a vertical asymptote. The division by zero kills continuity." By comparison a 2023 version of [http://math.columbia.edu/~avizeff/calculus-I-F23/lecture-5.pdf introductory calculus lecture notes] from Columbia have "However, this new definition doesn’t necessarily correlate with our intuition of which functions are continuous. For example, think about the function {{tmath|1=f(x) = \tfrac1x}}. This clearly has a discontinuity at {{tmath|0}}: the limit {{tmath|\textstyle \lim_{x\to0} \tfrac1x }} does not exist, and neither does {{tmath|f(0)}}, so we certainly can’t say that they’re equal if they’re not even defined. However, at every other point f(x) is continuous, and since 0 is not in its domain this means f(x) is continuous everywhere in its domain and therefore continuous!" –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 07:10, 13 April 2024 (UTC)
:::::::::::Uhm, I'm fine with discussing both points of view, but to me the current article looks "A, but some authors B" and I prefer "B, but some A". [[User:慈居|慈居]] ([[User talk:慈居|talk]]) 08:21, 13 April 2024 (UTC)
::::::::::::Repeating D.Lazard from above, the current article says explicitly:
::::::::::::{{tq|i=yes|"Many commonly encountered functions are [[partial function]]s that have a domain formed by all real numbers, except some [[isolated point]]s. Examples are the functions <math display="inline">x \mapsto \frac {1}{x}</math> and <math>x\mapsto \tan x.</math> When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are discontinuous."}}
::::::::::::What would be your preferred wording? –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 16:01, 13 April 2024 (UTC)
:::::::::::::If I may try (but in better English than mine): Many commonly encountered functions have a domain formed by all real numbers, except some isolated points. Examples are the functions <math>\textstyle x\mapsto\frac1x</math> and <math>x\mapsto\tan x</math>. They are continuous functions, since each of them is continuous at every point in the domain. They are not continuous nor discontinous at the exceptional points, since those points do not belong to the domain. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says that they are discontinuous at those points, and therefore not continuous functions on the entire real line. [[User:慈居|慈居]] ([[User talk:慈居|talk]]) 16:20, 13 April 2024 (UTC)
::::::::::::::I second that, except for the second part ("In other contexts ..."), which is what currently is in the article, vaguely phrased (when is one "interested in their behavior near exceptional points"?), and directly contradicting the very definition of discontinuity (e.g. for the epsilon-delta criterion, the point in question has to be part of the domain) and also some of the sources and the source I gave from a maths textbook (see above, Arens, Hettlich - Mathematik (2018), page 219), plus the French and German Wikipedia on this topic  --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 11:14, 15 April 2024 (UTC)
::::::::::::::I think this is a very good suggestion [[User:Math Michi|Math Michi]] ([[User talk:Math Michi|talk]]) 10:03, 4 September 2024 (UTC)
:::::::::::One of my reasons is that this should be at least more compatible with the definition for [[topological space]]s. There is no apparent natural parent spaces asggined to arbitrary spaces, so continuity of maps between spaces is only defined at points in the domain. [[User:慈居|慈居]] ([[User talk:慈居|talk]]) 08:27, 13 April 2024 (UTC)
:::::::'''Continuity''' is defined via the epsilon-delta criterion. A function is continuous if it is continuous at every point in its domain <math>D_f</math>. This means that the function satisfies the <math>\varepsilon</math>-<math>\delta</math> criterion at every point in its domain and is thus called a continuous function. However, this is not true for <math>\frac{1}{x}</math> at <math>x = 0</math>.
:::::::In the real numbers <math>\mathbb{R}</math>, continuity at a point is equivalent to the limit criterion (assuming you accept the axiom of choice, which is needed in the proof). The criterion, in a simplified form, states: a function <math>f</math> is continuous at <math>x_0</math> if <math>\lim_{{x \to x_0}} f(x) = f(x_0)</math>.
:::::::You are referring to the left-hand side not existing. However, the actual problem lies with the equality and the right-hand side, which ultimately leads to the same conclusion as the <math>\varepsilon</math>-<math>\delta</math> criterion. [[User:Math Michi|Math Michi]] ([[User talk:Math Michi|talk]]) 12:51, 4 September 2024 (UTC)
:::::This is gold. According to all variants you named except one, {{tmath| x \mapsto 1/x}} is still a continuous function - and via the only (niche and most of all, inaccurate) one that makes {{tmath| x \mapsto 1/x}} a discontinuous function, e.g. {{tmath| x \mapsto 1/x^2}} would be a continuous function, using the same criteria. Wow! Also, treating {{tmath| x \mapsto 1/x}} as a (non-total) partial function, there is no discontinuity at x=0, because, and I'm repeating myself here, x=0 is not part of the domain, so no criterion for discontinuity is applicable there. Treating it as a partial function doesn't change that in the slightest and it shouldn't matter whether there are authors who simply ignore this fact. Well, I'm out of here, both actively and passively, and you can even ban my account, I don't care any more.
:::::I aimed at improving this article by employing the logical rigor actually required (and special) to mathematics, and provided the according sources, but it had no effect and nobody came to my support (maybe because a professor is involved? Also, some editors with the same cause already gave up, which can be seen in the archives). Of course there are sources with less rigorous standards, so there are controversial or even self-contradictory terms like 'infinite discontinuity' or "multivalued function" to be found on the internet or even in papers and books. In our lab, we used to often say jokingly: "Well, it's published, therefor it has to be true (laughs)" - There are high-quality sources, low-quality sources and everything in between. So far, I was under the impression that the auto-correcting forces within Wikipedia, and especially for important and basic/essential mathematics articles like this one, would naturally shape them guided by sources that are to the largest extent of high quality, used by editors who try to work towards the best possible encyclopedic entry, instead of to push through their own, non-mainstream, ideas.
:::::But this is clearly not the case. Editors who have a personal interest in popularizing their own research and ideas, that are sometimes not even consistent with definitions in the same article they end up with, force these onto the masses by largely rewriting fundamental English Wikipedia articles like this one (the article 'Function (mathematics)' is another example) in a clearly biased fashion and with obvious issues, and ensuing valid criticism (not merely mine) on the talk-page is then answered with circular arguments (see talk-page of 'Multivalued function'), non-arguments (e.g. see my initial discussion with [[User:D.Lazard|D.Lazard]], where, in one response he described what 'poles' are, which was never in question and didn't help his argumentation at all), sarcasm ("congratulation, you will win the [[Millenium Prize Problems|Millenium Prize]] for having solved the [[Riemann hypothesis]]" - but only I am later called out for sarcasm) or the assertion that many reliable sources support the case, followed by the production of a citation that actually does the opposite! Pointing all of this out is simply ignored. Then the article is outright vandalized by adding almost completely unrelated complex number content in the section 'Real Functions', which only I objected to and nobody took action against for almost two years. That alone should be an indication for a lack of editors (not readers!) still caring for this article.
:::::In summary, none of this matters because the conflict is just "waited out" and for several reasons nobody jumps in to notice that the criticisms are actually valid, haven't been addressed by the author, and thus the article should be reverted (plus other clarifications added). Realizing recently how many other articles have been "improved" recently, I deem the situation unsalvageable and concede this fight against windmills. Btw, I really didn't expect to find mechanisms at work here that I would have a contextualized mostly with politics.
:::::Finally, as it is now, because this is the Wikipedia with the widest reach, masses of students and otherwise interested individuals learn here, every day, that the square root of a positive real number y is usually taken to be both the positive and the negative solution of the equation y=x^2, that partial functions are almost as important as (total) functions in mathematics in general (even though they are a rather niche concept and mostly relevant in computer science, [[User:D.Lazard|D.Lazard]]'s field of research) and that e.g. rational functions are usually considered discontinuous functions, even though the definitions in the same article show this is plain wrong, no matter whether one considers them partial or total functions or uses extensions to the real numbers as functional values. No arguments, neither mine nor from [[User:慈居|慈居]] had an impact on the situation (they withdrew their interest being content with saving the Wikipedia article in their language). Note that, with my arguments and citations at hand, I could have, according to the rules, just made changes to the article myself, but I wanted to avoid an edit war and I was honestly hoping up to now that finally several other voices, e.g. the ones that argued in favor of my case a few years ago (part of which gave up, similarly to how I do now), would make themselves heard, so any changes in favor of a rigorously logical version would be enduring.
:::::I don't expect an answer, and I wouldn't read it. --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk:Felix Tritschler|talk]]) 12:32, 16 April 2024 (UTC)
::::::{{!tqq|Then the article is outright vandalized by adding almost completely unrelated complex number content in the section 'Real Functions'}}: Just for the record, {{u|Felix Tritschler}} repeats this again and again although there is no mention of complex numbers in {{alink|Real functions}}. Should we deduce that he criticizes the article and its editors without having read it? [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:56, 16 April 2024 (UTC)
:::::::Can we all try to stay polite, please, even in the face of rudeness?
:::::::Felix Tritschler: being gratuitously condescending isn't selling your argument. Nor are sarcasm, hyperbole, mischaracterization of the article and other editors' comments, or bizarre false accusations.
:::::::{{tq|i=yes|I was under the impression that the auto-correcting forces within Wikipedia [...] would naturally shape them}} – Sorry to shatter your illusion, but no, this has never been true. Wikipedia has always been, and remains, full of disagreements/disputes, contradictions, frustration, and huge amounts of work remaining. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 16:36, 16 April 2024 (UTC)
::::::{{tq|i=yes|No arguments, neither mine nor from 慈居 had an impact on the situation.}} The situation was in the progress! It takes time for the community to reach a consensus. This will only slow it down. [[User:慈居|慈居]] ([[User talk:慈居|talk]]) 16:54, 16 April 2024 (UTC)
::::Then perhaps better use the standard definition (as in Starng's ''Calculus'') by default in the article? As for the other one, we could also mention that, citing a reference; I've seen such a definition, but not in an English reference, so I'm not going to add one myself. [[User:慈居|慈居]] ([[User talk:慈居|talk]]) 17:37, 12 April 2024 (UTC)

In both the real number system and the complex number system, infinity is not a number. In particular, the real number system obeys the Archimedean axiom. which says that for any two positive real numbers, some integer multiple of the smaller is bigger than the larger. If infinity were allowed as a real number, the Archimedean axiom would not hold.

What is called calculus at a lower level is, at a higher level, called either Real Analysis or Complex Analysis, the name indicating which number system is used. [[User:Rick Norwood|Rick Norwood]] ([[User talk:Rick Norwood|talk]]) 11:32, 28 October 2023 (UTC)

== Epsilon-Delta Definition? ==

An epsilon-delta definition of continuity is mentioned, but not explicitly given. Weierstrass gave one. We can cite Weierstrass and use the following:

<math>\forall \epsilon > 0:\exists \delta > 0:\forall x \in \reals:0<\left|x - a\right|<\delta\rightarrow\left|f(x) - f(a)\right|<\epsilon</math>

The epsilon-delta definition of continuity is used to define a notion of the limit based on continuous extension. [[User:TheodoricStier|TheodoricStier]] ([[User talk:TheodoricStier|talk]]) 14:29, 12 February 2025 (UTC)

:Did you read the article before posting here? There is a section {{alink|Weierstrass and Jordan definitions (epsilon–delta) of continuous functions}}, where this definition is explicitely given twice, with some prose included for an easier reading. Also, the lead povides an explicit link to the [[(ε, δ)-definition of limit]], which allows an easy recovering of your definition of continuity, by replacing "limit" by its (ε, δ)-definition in the definition of continuity in terms of limits. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:21, 12 February 2025 (UTC)
::Ah i can see it's been spelled out quite differently from how I'd expect it to appear while scanning. [[User:TheodoricStier|TheodoricStier]] ([[User talk:TheodoricStier|talk]]) 14:52, 15 February 2025 (UTC)