Talk:Semi-continuity

Description inconsistent with formal definition and figures swapped

As on today, the opening of the article describes the semi-continuities as below. In fact, are not much higher (respectively, lower) should be replaced by are not much lower (respectively, higher). Also, as on today the graphs of lower semi-continuous and upper semi-continuous functions are swapped. The figure showing the upper semi-continuous function at $x_{0}$ is lower semi-continuous and vice varsa.

In mathematical analysis, semi-continuity (or semicontinuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function $f$ is upper (respectively, lower) semi-continuous at a point $x_{0}$ if, roughly speaking, the function values for arguments near $x_{0}$ are not much higher (respectively, lower) than $f\left(x_{0}\right).$ Dr. Dinesh Karia^{(Talk) (contribs)} 13:05, 8 September 2021 (UTC)

I'll check with a math PhD friend tonight, but you're onto something. I think that actually the diagram and the definition in the Characterization section (which are the same) are correct, and the definition in Formal Definition and the Intro, which is different, is wrong. The definition in Formal Definition and Intro doesn't seem to me to make sense. It requires that in a neighborhood of a point x_0, f(x) > f(x_o), if I read it right, which would mean that a function was upper semicontinuous only at isolated point where it dipped discontinuously and then jumped back up. editeur24 (talk) 17:15, 8 September 2021 (UTC)

I read it wrong, so my objection in the last paragraph is invalid. editeur24 (talk) 20:25, 13 September 2021 (UTC)

Equivalence of definitions

I'm not sure the formal definition given is quite correct. For instance, consider the function $f:\mathbb {R} \to [-\infty ,\infty ]$ given by $f(x):=-1/x^{2}$ for $x\neq 0$ and $f(x):=-\infty$ for $x=0$. This is upper-semicontinuous at $0$ according to the limsup definition, but for any $\epsilon >0$ we have $f(0)+\epsilon =-\infty$ and on every neighbourhood of $0$ the function takes some finite values, which would mean that $f$ is not upper-semicontinuous at $0$ according to the definition given. If the definition instead reads "$f$ is upper-semicontinuous at $x_{0}$ if for every $a>f(x_{0})$ there is a neighbourhood $U$ of $x_{0}$ such that $f(x)<a$ for all $x\in U$", then I think this is equivalent to the limsup definition. Note that if $f(x_{0})=\infty$ then the above condition is satisfied vacuously. 130.56.234.55 (talk) 01:10, 21 June 2013 (UTC)

You are correct, the $\epsilon$-neighborhood definition (for upper semi-continuity) requires that the point $x_{0}\in \{x\in X:f(x)>-\infty \}$, otherwise it fails (as your example illustrates). Zfeinst (talk) 17:51, 21 June 2013 (UTC)

First-countability

I suppose that the equivalences described in the "Formal definition"-section are equivalences only, if X is first-contable, since continuity always implies sequence-continuity, but the other direction being true only in first-countable spaces. Cf. Munkers: Topology, p. 190ff, or cf. the German Wikipedia article on "Halbstetigkeit", where X is required to be a metric space, which implies first-countability. Luzern66 (talk) 14:58, 3 July 2011 (UTC)

This definition covers only R and not R^n

This definition only covers functions with $f:X\rightarrow \mathbb {R}$. The definition for $f:X\rightarrow \mathbb {R} ^{n}$ is important for fixed point theorems. --Clausen 23:31, 26 May 2005 (UTC)

I would challenge you to add a section at the bottom talking about R^n. :) This might be a better option than rewriting all the article in the general R^n case. Will you take the challenge? :) Oleg Alexandrov 00:25, 27 May 2005 (UTC)

Lower Semi-continous Example

I couldn't make any sense of this:

Imagine that you are scanning a certain scenery with your eyes and record the distance to the viewed      
object at all times. This yields a lower semi-continuous function which in general is not upper 
semi-continuous (for instance if you focus on the edge of a table).

Would someone like to clarify it? Or else we could just use a conventional example like $f(x)=\lceil x\rceil$ Deepak 16:17, 31 March 2006 (UTC)

• Ok then, I'm going to make the change. Deepak 16:52, 20 April 2006 (UTC)

Are we sure about the lower semi-continuous definition in terms of having a neighborhood around x_0 where all the x in the nbhd have f(x) > f(x_0) ? given that we are also saying that a continuous function is both upper and lower semi-continuous, then taking the identity function and fixing any point should give us a l.s.c. function where any neighborhood will have some x with f(x1) < f(x0) < f(x2) where nbhd is fixed around x0, x1, x2 are in the nbhd. Maybe just use the lim inf definition.

18:12, 8 October 2006 (UTC)chuck

I think the article is correct as it stands: the neighbourhood definition does not state that f(x) > f(x_0) in the neighbourhood, but rather states that f(x) > f(x_0) - ε, and this would fit OK with continuous functions such as the identity function. Madmath789 21:50, 8 October 2006 (UTC)

Order topology

Upper (resp. lower) semi-continuity is continuity with respect to the right (resp. left) order topology on R. Is there any point mentioning this here? Algebraist 12:14, 24 March 2008 (UTC)

Not sure. That seems to be pretty esoteric stuff, and semi-continuity is already complicated enough for most readers, even with the ordinary topology. Oleg Alexandrov (talk) 15:04, 24 March 2008 (UTC)

Fair enough. That saves me the effort of finding a source, anyway. Algebraist 18:32, 28 March 2008 (UTC)

Are you sure? the order topology of R is equivalent to the standard one. (Anonymous)

I have now added this in the definitions section. It's really equivalent to some of the other characterizations, and conceptually useful in some contexts. PatrickR2 (talk) 09:07, 17 November 2021 (UTC)

The figure may involve confusion

The figure may involve a confusion with cadlag functions, it should be completed to avoid this confusion —Preceding unsigned comment added by 92.141.19.110 (talk) 13:40, 22 April 2010 (UTC)

You show the same graph with the same point x_0 as being both upper and lower semi-continuous. That would mean the function in the picture is continuous, which isn't. Could you please clarify? —Preceding unsigned comment added by Serban00 (talk • contribs) 01:11, 2 December 2010 (UTC)

The functions in the two graphs are almost the same, but they're not identical. On one graph, the open point is at the top of the jump discontinuity, and on the other, it's at the bottom. The open point indicates that the function does not exist at that point for x₀, but does exist at the neighboring points up to that open point on one side; the open point is therefore the limit of f(x) as x approaches x₀ from that one side. Both functions are called semi-continuous because f(x) exists on the neighboring points of f(x₀), but on only one side instead of both. Mktyscn (talk) 05:08, 4 December 2010 (UTC)

Merge discussion (closed convex function)

I think the page on closed convex functions should be merged with semi-continuity since a closed convex function is merely a lower semi-continuous and convex function. What does everyone else think? Zfeinst (talk) 04:08, 28 May 2012 (UTC)

I disagree, because convex functions can be allowed to take the values +infinity and -infinity, and this introduces complications your reduction cannot capture. Some authors will call a convex function closed if it coincides with the pointwise supremum of its affine minorants; others require instead that every sub-level set be closed. When the value -infinity is allowed, these criteria are different. Perhaps the page on closed convex functions could be refined, but not in the suggested manner. — Preceding unsigned comment added by 137.82.36.82 (talk) 18:04, 4 July 2012 (UTC)

I think adding material on convex functions to this article would just make it more confusing. There is also probably ample scope for a separate article on closed convex functions. That article needs more explanations, references, and examples. Once those are added, I think it will be clear that these are distinct topics. Sławomir Biały (talk) 14:30, 23 February 2013 (UTC)

Topology of almost everywhere convergence?

In the examples section, it says that

Let $(X,\mu )$ be a measure space and let $L^{+}(X,\mu )$ denote the set of positive measurable functions endowed with the topology of $\mu$-almost everywhere convergence. Then the integral, seen as an operator from $L^{+}(X,\mu )$ to $[-\infty ,+\infty ]$ is lower semi-continuous. This is just Fatou's lemma.

I like the example, but it is false and I don't know a neat way to fix it. It is known that the notion of almost everywhere convergence doesn't come from a topology (unlike the notion of pointwise (everywhere) convergence, which comes from the product topology). If we use the topology of pointwise convergence (which would require not identifying two functions that are almost everywhere equal, or more easily, restricting to continuous functions), then the statement is well-formed but false.

How is this possible? Fatou's lemma talks about sequences and it is known that the convergent sequences in a topological space do not characterize its topology completely. A basic neighborhood of a function $f$ in the topology of pointwise convergence is a set of functions which nearly coincide with $f$ in a certain (finite) set of points, and this information is not enough to say anything about the values of the integrals of the functions. So all we could say is that integration is sequentially lower continuous. Marcosaedro (talk) 19:42, 27 January 2013 (UTC)

I have a thought about how to possibly repair this. Replace the "topology" on the space by that of convergence in measure. This comes from a metrizable topology, so there is no awkwardness in saying the "topology of convergence in measure". Fatou's theorem also holds for sequences that converge in measure, with essentially the same proof. (Indeed, Fatou follows straightforwardly from Lebesgue's monotone convergence theorem. For monotone sequences of functions, there is no difference between convergence in measure and convergence a.e., ... QED). Sławomir Biały (talk) 22:07, 22 February 2013 (UTC)

Some point-wise limits of lower semicontinuous functions.

Can the 'Semicontinuity' article be improved by adding the following (what appears to be) basic fact?

Fact. Every lower semicontinuous function f can be obtained as a point-wise limit from above of a monotone sequence fn of finite-valued lower semicontinuous functions.

Is this fact trivial/false/unknown etc? Any examples/counter-examples? Any references? Aucross (talk) 03:59, 5 February 2015 (UTC)

Suggestion for slight improvement

The text says "Multiplying a positive upper semi-continuous function with a negative number turns it into a lower semi-continuous function." But there seems to be no need to ask that the upper semi-continuous function is positive. We can actually deduce this from two useful facts: 1) $f$ is lsc iff $-f$ is usc because $-f(x)<\alpha$ iff $f(x)>-\alpha$ 2) multiplying by a positive real number preserves lsc and usc.

Streamlining of the definition section

The Formal Definition section of the article has become unnecessarily WP:TECHNICAL and should be simplified. Browsing at the history of the article, it appears that most of the changes introducing these technicalities were added by user Mgkrupa. While some of his changes improved readability (for example having separate subsections for upper and lower semicontinuity, and the reordering of some of the sections), the additional minutiae made the new article harder than it needs to be. If other people agree with this, I will do the change.

In this particular case, the pre-Mgkrupa version of the definition (https://en.wikipedia.org/w/index.php?title=Semi-continuity&oldid=959632505) was correct, complete and to the point (for every $y>f(x_{0})$ there is a neighborhood $U$ such that ...). Mgkrupa added several paragraphs of considerations about the case of $+\infty$ and "separating into two cases", etc, which seems totally unnecessary. While there is nothing logically incorrect about these paragraphs, they don't add anything to the understanding of the concept, and more importantly, they are all logical consequences of the definition anyway. These minutiae dilute the impact of the article and make it unnecessarily technical. They also go against WP:NOTTEXTBOOK. Sometimes "less is more" :-) PatrickR2 (talk) 23:56, 12 November 2021 (UTC)

I removed some of that info. Let me know if you want more changes. And feel free to make the changes yourself.Mgkrupa 00:43, 13 November 2021 (UTC)

Thank you Mgkrupa, this is a lot more readable. I appreciate that change. PatrickR2 (talk) 00:47, 13 November 2021 (UTC)

Good to see a nice polite interchange like that. Good for you guys. editeur24 (talk) 02:49, 13 November 2021 (UTC)

Semi-continuity versus semicontinuity

I am curious why all instances of "semicontinous" without a hyphen have been changed to the version with a hyphen. All the (pure mathematics) topology and analysis books that I have (Willard, Engelking, Royden, Stromberg, etc) seem to use the version without a hyphen. Same thing in the SIAM Review article in the Bibliography section of the article, and in https://encyclopediaofmath.org/wiki/Semicontinuous_function.

Also a search in both https://math.stackexchange.com and https://mathoverflow.net shows "semicontinuous" to be more commonly used. In addition the following comparison in Google books between "lower semicontinuous" and "lower semi-continuous" shows the version without a hyphen is about five times more common in recent books:

https://books.google.com/ngrams/graph?content=lower+semi-continuous%2Clower+semicontinuous&year_start=1910&year_end=2019&corpus=26&smoothing=3&direct_url=t1%3B%2Clower%20semi%20-%20continuous%3B%2Cc0%3B.t1%3B%2Clower%20semicontinuous%3B%2Cc0#t1%3B%2Clower%20semi%20-%20continuous%3B%2Cc0%3B.t1%3B%2Clower%20semicontinuous%3B%2Cc0

The version without a hyphen is also easier to type. PatrickR2 (talk) 16:36, 15 November 2021 (UTC)

Same type of results for searches in https://arxiv.org/search/math. The version without hyphen is used about fives times more commonly in academic research in mathematics. PatrickR2 (talk) 16:52, 15 November 2021 (UTC)

@Editeur24: Would it be acceptable to you if I change the article to use the version without the hyphen? PatrickR2 (talk) 23:50, 16 November 2021 (UTC)

How nice of you to ask! Yes, it looks like people have checked it out. I think I like it better without the hyphen too, actually, just from the point of view of omitting a needless character. I think I may have asked a math professor friend with strong opinions who liked the hyphen, but go ahead and take it out.editeur24 (talk) 00:52, 17 November 2021 (UTC)

Thanks. I'll make the edit. PatrickR2 (talk) 01:09, 17 November 2021 (UTC)

Based on this discussion I propose moving the page from "Semi-continuity" to "Semicontinuity", to reflect the usage throughout the article. The-erinaceous-one (talk) 19:31, 8 August 2024 (UTC)

Sounds good to me. PatrickR2 (talk) 00:58, 9 August 2024 (UTC)

It looks like it might be nontrivial to move the page, since "Semicontinuity" already exists as a redirect with multiple edits, so we may need an admin to make the move.

Anyway, before making the move, perhaps "Semicontinuous function" would be preferable to "Semicontinuity", to match the naming of the article Continuous function? The-erinaceous-one (talk) 08:34, 9 August 2024 (UTC)

I am not sure myself about all the steps involved (including all the related redirect) for such a move, but I am sure there are administrative tools out there. I think there are special requests to be made and then discussed, before things get finalized by an admin. But don't remember right now how that works. Worth asking on https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics (WPM).

About "semicontinuity" versus "semicontinuous function", either one seems ok to me. That's another thing that can be discussed in WPM. PatrickR2 (talk) 03:47, 10 August 2024 (UTC)

On the other hand, we also have "hemicontinuity". So maybe "semicontinuity" is fine. PatrickR2 (talk) 03:49, 10 August 2024 (UTC)

Unrelated, but did you see my last comment in the section "Hemicontinuity does not belong here"? PatrickR2 (talk) 03:48, 10 August 2024 (UTC)

Semicontinuous function as limit of monotone sequence of continuous functions

The article says "Every lower semi-continuous function on a uniform space (e.g. a metric space) arises as the supremum of a sequence of continuous functions". This is true when the domain is a metric space. But I don't think it is correct for uniform spaces in general. A topological space is uniformizable exactly when it is completely regular. On the other hand, it was proved by Hing Tong in 1952 that a topological space has the property that any lower semicontinuous function is the limit of a monotonically increasing sequence of continuous functions exactly when the space is perfectly normal. See Engelking exercise 1.7.15(c) for a sketch of the proof, also reproduced in https://math.stackexchange.com/questions/1279763. Since completely regular spaces are not even normal in general, the result cannot be true for uniform spaces. PatrickR2 (talk) 17:27, 15 November 2021 (UTC)

I have now fixed this. PatrickR2 (talk) 17:49, 15 November 2021 (UTC)

equivalences in the section on lower semicontinuity

In my understanding of the section on lower semicontinuity, condition (1) is not equivalent with the others, for instance with (3) (analogously the section on upper semicontinuity). Take the one-dimensional example f(x)=0 if x>0 and f(x)= +infinity else. Then, f is lower semicontinuous (actually: continuous) at every point of its domain (positive reals), but the sublevel set with respect to the level 0 (or any level y larger than zero) equals its domain which is open but not closed. 62.141.176.2 (talk) 15:11, 24 November 2022 (UTC)

I assume that you want the function $f$ in your example to be defined on the space $X=\mathbb {R}$. That function is not lower semicontinuous at $x=0$. And of course it's not continuous at that point either. Note that the domain is all of $\mathbb {R}$ and not just the set of positive reals. PatrickR2 (talk) 02:22, 26 November 2022 (UTC)

Hemicontinuity does not belong here

@The-erinaceous-one: You have recently added a section about semicontinuity of set-valued functions, as well as mentioned upper, lower, inner, outer variants without any explanation. All this seems misguided, as the current article is focused on (extended) real-valued functions. Like you yourself mentioned, for set-valued functions there is the Hemicontinuity article, where details about the set-valued equivalents can be expanded at length. On the other hand, the "See also" section can have a link to Hemicontinuity, which it actually already have in a disguised form. (I'll change it shortly to make it clearer.) That is exactly the purpose of the See also section, see MOS:SEEALSO: One purpose of "See also" links is to enable readers to explore tangentially related topics.

So I propose to remove the section on semicontinuity of set-valued functions from this article. Comments? PatrickR2 (talk) 21:33, 1 August 2024 (UTC)

Furthermore, semicontinuity of set-valued functions with a link to Hemicontinuity is even mentioned in a hat note right at the top of the article. Even more reason to eliminate your new section from here. PatrickR2 (talk) 21:39, 1 August 2024 (UTC)

There is a strong connection between upper semicontinuous set-valued functions and upper semicontinuous single-valued functions, but I didn't have sufficient time to research it and write it out. Rather than removing the section, it would be better to expand it and explain how they are related and how they differ.

There needs to be a broader discussion about the organization of the topics of semicontinuous single-valued functions, semicontinuous set-valued functions, and hemicontinuous set-valued functions. If we exclude set-valued semicontinuity from semicontinuity, then would either need to showhorn it into hemicontinuity, or create a new set-valued semicontinuity page. Both of those seem like bad alternatives. I would prefer to make semicontinuity include both set-valued and single-valued semicontinuity, with hemicontinuity constrained to the rather niche topic of hemicontinuity. I'm open to suggestions for alternatives, however. The-erinaceous-one (talk) 23:40, 1 August 2024 (UTC)

Right now, the current article is focused on real valued functions. The whole article, including the Examples and Properties section is tightly organized based on that and already contains a fair amount of information. Adding and mixing in definitions and results about set-valued functions would make the whole thing harder to read and to grasp, especially to most people who are more interested in the common real valued case. So I will delete the Definitions subsection you added for now.

If we want to add information about set-valued functions, it should be in a separate section titled for example "Generalization to set-valued functions" after the other sections. That would keep the whole article better organized and more accessible to the average reader. Or alternatively in a separate article. (A suggestion: first write a page in your sandbox space with what you want to add, so you better see the best way to organize things and we don't get half baked information in the current article.)

By the way, I was not familiar with hemicontinuity until now, but that article seems it needs improvement. For example, the notation $\Gamma :A\rightrightarrows B$ is never explained. It should also mention the relationship between hemicontinuity, semicontinuity and continuity. And Set-valued function is worse, explained in a rather fuzzy and unclear way with a lot to be desired. PatrickR2 (talk) 05:52, 2 August 2024 (UTC)

@The-erinaceous-one: Much better organized now, with the set-valued stuff in its own section. One thing is unclear: In the definition of lower semicontinuity, what does $F^{-1}(U)$ even mean for a set-valued function? This is not explained in Set-valued function either. PatrickR2 (talk) 06:53, 8 August 2024 (UTC)

I have a definition at the top of the set-valued section. Is it unclear? The-erinaceous-one (talk) 07:41, 13 August 2024 (UTC)

Yeah, I see it now. I must have missed it somehow, as it was not there originally. By the way, it is not phrased correctly: it should be the preimage of a set under a function, and not the preimage of a function under a set! Also, when adding a reference to a big book, it is more helpful to readers to mention a specific section or page. For this kind of things, it's preferable to add the book itself (like the one of Freeman & Kotokovic) in the Bibliography section and then have the reference be something like {{sfn|Freeman|Kotokovic|...|p (or loc)=...}} for example. See the references to Willard or Stromberg for example. (I did not use sfn as I did not know about it then, but at least the specific section/page is separated from the full ref for the book.) PatrickR2 (talk) 18:37, 13 August 2024 (UTC)

I see that you use the rp template. That works too. In any case, we need a page/section number for this definition. Or atlternatively, don't define it here, but define it in the article Set-valued function? PatrickR2 (talk) 19:08, 13 August 2024 (UTC)

Huh, I haven't seen the sfn template before. I'll look into using it. The formatting seems better than the rp template. The-erinaceous-one (talk) 08:11, 14 August 2024 (UTC)

Problems with latest edits (Aug 2024)

@The-erinaceous-one: Sorry about reverting a few of your changes (I actually intend to revert a few more things, with explanations and discussion here). But let's start with the section "Semicontinuity in Metric Spaces". That seems very redundant with what was already there. Furthermore, it is not even mathematically correct, and it has a few violations of MOS:MATH. What was your purpose in adding that section? PatrickR2 (talk) 03:37, 15 August 2024 (UTC)

I disagree regarding the redundancy. Perhaps we could move it out of the "definitions" section, but I think that including it (somewhere) helps put the concept of semicontinuity into a more concrete setting that most people would find easier to understand. The-erinaceous-one (talk) 21:12, 15 August 2024 (UTC)

You'll need to be more specific for "not mathematically correct" and "has a few violations of MOS:MATH". Vague comments like that are not constructive or helpful. The-erinaceous-one (talk) 21:27, 15 August 2024 (UTC)

I'll reply to this specifically later. In the mean time, let me comment separately below on the Theorem of Baire. PatrickR2 (talk) 01:18, 16 August 2024 (UTC)

@The-erinaceous-one: What I was referring to is the use of the quantifier symbol $\forall x...$. MOS:MATH tells us to use the English equivalent for that. As for the mathematical mistake, for upper semicontinuity for example, suppose $f(x_{0})=-\infty$. Then $f(x_{0})+\varepsilon$ also equals $-\infty$, so the definition as you had it would mean that $f(x)$ would need to be constant with value $-\infty$ in a whole neighborhood of $x_{0}$, which is not correct. I don't have access to Aubin's book, but if that's taken from that book, that reference is really sloppy and one wonders if it can be trusted.

Anyway, it's fine if we want to add something about this "more concrete setting". But I think it would be preferable to have this directly next to the general definition for ease of comparison. I'll do the modification, in a slightly more streamlined way. If you later disagree with my change, please discuss here first instead of reverting. PatrickR2 (talk) 20:31, 21 August 2024 (UTC)

Regarding the Theorem of Baire: Thank you for removing the "theorem box" for this; it looks much better. From the changes that you have made recently to this article, and to other articles as well, it seems you are a fan of display mode and theorem boxes and proof boxes. They have their place sometimes (for example in an article or section dedicated to a specific theorem), but at other times they are misguided. The main problem with them is that they give undue weight to the particular result being highlighted (WP:WEIGHT). In this case, the theorem of Baire is a nice result, but not more important that some of the other results in the Properties section. For example, I would personally view the converse result that the limit of an increasing sequence of continuous functions is lower semicontinuous to be even more important. So not using a "theorem box" is preferable in this case.

And as we are on the topic, I earlier removed the "proof box" for the other result for a similar reason. We don't want to put too much emphasis on proofs. This has been discussed repeatedly on WPM. See also the Proofs section of MOS:MATH: A downside of including proofs is that they may interrupt the flow of the article, whose goal is usually expository. Use your judgment; as a rule of thumb, include proofs when they expose or illuminate the concept or idea; don't include them when they serve only to establish the correctness of a result. (This is the reason I included the proof for that particular result and no other.) And regarding the use of math display mode, sometimes it's helpful, sometimes it's unnecessary and misguided and breaking the flow of other things, and giving undue weight to some auxiliary notation or equation. Anyway, hope that clarifies my reasoning for all this. PatrickR2 (talk) 01:52, 16 August 2024 (UTC)

@The-erinaceous-one I'd be interested to hear your views on this. PatrickR2 (talk) 03:44, 16 August 2024 (UTC)

I'm fine with not using {{Math theorem}} and{{Math proof}}. They are a little overbearing. I think it would be cool to add options to those templates to allow configurations that would make them more suitble for wide use, such as removing the box and displaying proofs in a collapsed environment, so that they are hidden by default, but that is a discussion for another place. The-erinaceous-one (talk) 05:59, 18 August 2024 (UTC)

Regarding the lattice property: @The-erinaceous-one The fact that lower semicontinuous functions form a lattice is intimately tied to the fact that the max and min preserve such functions. One could basically view it as a restatement of that fact. So it is preferable to keep the two together. I have made the change. (Also a minor thing: according to Wikipedia guidelines, there should be no blank lines between items in a list; I have cleaned that as well.) PatrickR2 (talk) 04:46, 25 August 2024 (UTC)

@PatrickR2: That's fine with me. I separated them because I found the way it was initially stated to be a bit confusing, but the current form looks better. The-erinaceous-one (talk) 21:50, 25 August 2024 (UTC)