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1862:. 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).
22:
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3289:). 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.
3302:(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.
2234:
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
3343:. 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.
2389:
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
3367:
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
3284:
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
2965:
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
2299:
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
2212:
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
2048:. 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.
1549:
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
1315:
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,
2748:
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
2709:
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
1794:
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
2143:
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
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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
3263:
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,
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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
2459:
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
367:
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
2888:
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
1123:
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
2656:
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
1909:
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)
1250:
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).
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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.
2985:
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.
2228:
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:
2713:
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
3292:
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
2889:
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
2515:
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:
2753:
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.)
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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.)
1798:
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
1795:
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.
2749:
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.
1463:
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?
1232:
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? :)
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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
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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
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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 (
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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
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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.
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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.
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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.
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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
2665:, 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
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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
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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
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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.
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Not sure. That seems to be pretty esoteric stuff, and semi-continuity is already complicated enough for most readers, even with the ordinary topology.
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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
1580:. That article needs more explanations, references, and examples. Once those are added, I think it will be clear that these are distinct topics.
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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
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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.
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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) : -->
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Anyway, before making the move, perhaps "Semicontinuous function" would be preferable to "Semicontinuity", to match the naming of the article
1499:, 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 x
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Based on this discussion I propose moving the page from "Semi-continuity" to "Semicontinuity", to reflect the usage throughout the article.
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About "semicontinuity" versus "semicontinuous function", either one seems ok to me. That's another thing that can be discussed in WPM.
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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
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tells us to use the
English equivalent for that. As for the mathematical mistake, for upper semicontinuity for example, suppose
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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) : -->
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is even mentioned in a hat note right at the top of the article. Even more reason to eliminate your new section from here.
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that every sub-level set be closed. When the value -infinity is allowed, these criteria are different. Perhaps the page on
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is never explained. It should also mention the relationship between hemicontinuity, semicontinuity and continuity. And
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Huh, I haven't seen the sfn template before. I'll look into using it. The formatting seems better than the rp template.
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since a closed convex function is merely a lower semi-continuous and convex function. What does everyone else think?
1495:, and on the other, it's at the bottom. The open point indicates that the function does not exist at that point for x
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I'll reply to this specifically later. In the mean time, let me comment separately below on the
Theorem of Baire.
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Knowledge article on "Halbstetigkeit", where X is required to be a metric space, which implies first-countability.
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from that one side. Both functions are called semi-continuous because f(x) exists on the neighboring points of f(x
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2480:. Since completely regular spaces are not even normal in general, the result cannot be true for uniform spaces.
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to
Knowledge guidelines, there should be no blank lines between items in a list; I have cleaned that as well.)
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I removed some of that info. Let me know if you want more changes. And feel free to make the changes yourself.
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constrained to the rather niche topic of hemicontinuity. I'm open to suggestions for alternatives, however.
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So I propose to remove the section on semicontinuity of set-valued functions from this article. Comments?
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2258:. The version without hyphen is used about fives times more commonly in academic research in mathematics.
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The figure may involve a confusion with cadlag functions, it should be completed to avoid this confusion
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Can the 'Semicontinuity' article be improved by adding the following (what appears to be) basic fact?
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I have a thought about how to possibly repair this. Replace the "topology" on the space by that of
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It looks like it might be nontrivial to move the page, since "Semicontinuity" already exists as a
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Unrelated, but did you see my last comment in the section "Hemicontinuity does not belong here"?
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I'm not sure the formal definition given is quite correct. For instance, consider the function
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I think the article is correct as it stands: the neighbourhood definition does not state that
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Would it be acceptable to you if I change the article to use the version without the hyphen?
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One purpose of "See also" links is to enable readers to explore tangentially related topics
2607:. And of course it's not continuous at that point either. Note that the domain is all of
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dilute the impact of the article and make it unnecessarily technical. They also go against
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As on today, the opening of the article describes the semi-continuities as below. In fact,
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in the topology of pointwise convergence is a set of functions which nearly coincide with
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Upper (resp. lower) semi-continuity is continuity with respect to the right (resp. left)
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Is this fact trivial/false/unknown etc? Any examples/counter-examples? Any references?
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_0) - ε, and this would fit OK with continuous functions such as the identity function.
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Would someone like to clarify it? Or else we could just use a conventional example like
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On the other hand, we also have "hemicontinuity". So maybe "semicontinuity" is fine.
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such that ...). Mgkrupa added several paragraphs of considerations about the case of
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until now, but that article seems it needs improvement. For example, the notation
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2476:. See Engelking exercise 1.7.15(c) for a sketch of the proof, also reproduced in
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is worse, explained in a rather fuzzy and unclear way with a lot to be desired.
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Semicontinuous function as limit of monotone sequence of continuous functions
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denote the set of positive measurable functions endowed with the topology of
1031:-neighborhood definition (for upper semi-continuity) requires that the point
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1703:-almost everywhere convergence. Then the integral, seen as an operator from
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Thank you Mgkrupa, this is a lot more readable. I appreciate that change.
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https://en.wikipedia.org/search/?title=Semi-continuity&oldid=959632505
954:", then I think this is equivalent to the limsup definition. Note that if
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I have a definition at the top of the set-valued section. Is it unclear?
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The Formal Definition section of the article has become unnecessarily
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Good to see a nice polite interchange like that. Good for you guys.
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In this particular case, the pre-Mgkrupa version of the definition (
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according to the definition given. If the definition instead reads "
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Furthermore, semicontinuity of set-valued functions with a link to
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Fair enough. That saves me the effort of finding a source, anyway.
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I read it wrong, so my objection in the last paragraph is invalid.
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Description inconsistent with formal definition and figures swapped
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2) multiplying by a positive real number preserves lsc and usc.
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x1, x2 are in the nbhd. Maybe just use the lim inf definition.
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even mean for a set-valued function? This is not explained in
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include both set-valued and single-valued semicontinuity, with
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https://en.wikipedia.org/Wikipedia_talk:WikiProject_Mathematics
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the function takes some finite values, which would mean that
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if, roughly speaking, the function values for arguments near
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What I was referring to is the use of the quantifier symbol
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article in the Bibliography section of the article, and in
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Some point-wise limits of lower semicontinuous functions.
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https://encyclopediaofmath.org/Semicontinuous_function
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The version without a hyphen is also easier to type.
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equivalences in the section on lower semicontinuity
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1339:_0) in the neighbourhood, but rather states that
996:then the above condition is satisfied vacuously.
3264:please discuss here first instead of reverting.
2478:https://math.stackexchange.com/questions/1279763
621:according to the limsup definition, but for any
2581:. That function is not lower semicontinuous at
1211:{\displaystyle f:X\rightarrow \mathbb {R} ^{n}}
313:are not much higher (respectively, lower) than
3321:I'd be interested to hear your views on this.
1415:is equivalent to the standard one. (Anonymous)
2718:, then would either need to showhorn it into
8:
2891:{{sfn|Freeman|Kotokovic|...|p (or loc)=...}}
2785:{\displaystyle \Gamma :A\rightrightarrows B}
1282:
1276:
1088:
1052:
2553:in your example to be defined on the space
2368:, so we may need an admin to make the move.
1378:. Is there any point mentioning this here?
1170:{\displaystyle f:X\rightarrow \mathbb {R} }
1143:This definition only covers functions with
19:
1598:Topology of almost everywhere convergence?
1094:{\displaystyle x_{0}\in \{x\in X:f(x): -->
171:are not much lower (respectively, higher).
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1523:Merge discussion (closed convex function)
1507:), but on only one side instead of both.
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1139:This definition covers only R and not R^n
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167:are not much higher (respectively, lower)
2966:correct, and it has a few violations of
2629:and not just the set of positive reals.
2468:. On the other hand, it was proved by
1783:is lower semi-continuous. This is just
1602:In the examples section, it says that
1304:Ok then, I'm going to make the change.
687:{\displaystyle f(0)+\epsilon =-\infty }
49:
3298:
2670:
2036:Streamlining of the definition section
3210:would need to be constant with value
3151:{\displaystyle f(x_{0})+\varepsilon }
2954:Problems with latest edits (Aug 2024)
2254:Same type of results for searches in
2208:Semi-continuity versus semicontinuity
7:
2756:By the way, I was not familiar with
2533:I assume that you want the function
1411:Are you sure? the order topology of
1288:{\displaystyle f(x)=\lceil x\rceil }
347:{\displaystyle f\left(x_{0}\right).}
95:This article is within the scope of
2645:Hemicontinuity does not belong here
1247:I couldn't make any sense of this:
227:. An extended real-valued function
38:It is of interest to the following
3220:
3168:
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2657:set-valued functions there is the
2130:
1767:
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1085:
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601:. This is upper-semicontinuous at
562:
451:{\displaystyle f:\mathbb {R} \to }
442:
436:
14:
3420:Mid-priority mathematics articles
3109:{\displaystyle f(x_{0})=-\infty }
1905:Suggestion for slight improvement
1319:18:12, 8 October 2006 (UTC)chuck
115:Knowledge:WikiProject Mathematics
3415:Start-Class mathematics articles
1984:{\displaystyle -f(x)<\alpha }
1438:The figure may involve confusion
989:{\displaystyle f(x_{0})=\infty }
118:Template:WikiProject Mathematics
82:
72:
51:
20:
3361:Regarding the lattice property:
3282:Regarding the Theorem of Baire:
2148:. Sometimes "less is more" :-)
734:is not upper-semicontinuous at
135:This article has been rated as
3197:
3191:
3139:
3126:
3094:
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2843:
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2724:set-valued semicontinuity page
2574:{\displaystyle X=\mathbb {R} }
2222:https://math.stackexchange.com
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568:{\displaystyle f(x):=-\infty }
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504:{\displaystyle f(x):=-1/x^{2}}
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397:20:25, 13 September 2021 (UTC)
1:
2639:02:22, 26 November 2022 (UTC)
2528:15:11, 24 November 2022 (UTC)
2505:17:49, 15 November 2021 (UTC)
2490:17:27, 15 November 2021 (UTC)
2390:that works. Worth asking on
2332:01:09, 17 November 2021 (UTC)
2310:00:52, 17 November 2021 (UTC)
2291:23:50, 16 November 2021 (UTC)
2268:16:52, 15 November 2021 (UTC)
2256:https://arxiv.org/search/math
2249:16:36, 15 November 2021 (UTC)
2198:02:49, 13 November 2021 (UTC)
2184:00:47, 13 November 2021 (UTC)
2170:00:43, 13 November 2021 (UTC)
2158:23:56, 12 November 2021 (UTC)
1874:22:07, 22 February 2013 (UTC)
1738:{\displaystyle L^{+}(X,\mu )}
1676:{\displaystyle L^{+}(X,\mu )}
1592:14:30, 23 February 2013 (UTC)
1432:09:07, 17 November 2021 (UTC)
641:{\displaystyle \epsilon : -->
379:17:15, 8 September 2021 (UTC)
362:13:05, 8 September 2021 (UTC)
109:and see a list of open tasks.
3060:{\displaystyle \forall x...}
2622:{\displaystyle \mathbb {R} }
2366:redirect with multiple edits
2322:Thanks. I'll make the edit.
1900:03:59, 5 February 2015 (UTC)
1853:19:42, 27 January 2013 (UTC)
1517:05:08, 4 December 2010 (UTC)
1484:01:11, 2 December 2010 (UTC)
1243:Lower Semi-continous Example
3400:21:50, 25 August 2024 (UTC)
3378:04:46, 25 August 2024 (UTC)
3353:05:59, 18 August 2024 (UTC)
3331:03:44, 16 August 2024 (UTC)
3312:01:52, 16 August 2024 (UTC)
3274:20:31, 21 August 2024 (UTC)
3233:in a whole neighborhood of
3024:01:18, 16 August 2024 (UTC)
3010:21:27, 15 August 2024 (UTC)
2996:21:12, 15 August 2024 (UTC)
2980:03:37, 15 August 2024 (UTC)
2936:08:11, 14 August 2024 (UTC)
2922:19:08, 13 August 2024 (UTC)
2904:18:37, 13 August 2024 (UTC)
2884:07:41, 13 August 2024 (UTC)
2437:03:48, 10 August 2024 (UTC)
2421:03:49, 10 August 2024 (UTC)
2407:03:47, 10 August 2024 (UTC)
1641:be a measure space and let
1360:21:50, 8 October 2006 (UTC)
774:is upper-semicontinuous at
3436:
2870:06:53, 8 August 2024 (UTC)
2806:05:52, 2 August 2024 (UTC)
2744:23:40, 1 August 2024 (UTC)
2705:21:39, 1 August 2024 (UTC)
2686:21:33, 1 August 2024 (UTC)
2385:08:34, 9 August 2024 (UTC)
2360:00:58, 9 August 2024 (UTC)
2346:19:31, 8 August 2024 (UTC)
1458:13:40, 22 April 2010 (UTC)
1407:18:32, 28 March 2008 (UTC)
1398:15:04, 24 March 2008 (UTC)
1383:12:14, 24 March 2008 (UTC)
1309:16:52, 20 April 2006 (UTC)
1299:16:17, 31 March 2006 (UTC)
405:Equivalence of definitions
2849:{\displaystyle F^{-1}(U)}
1113:17:51, 21 June 2013 (UTC)
1024:{\displaystyle \epsilon }
1006:01:10, 21 June 2013 (UTC)
921:{\displaystyle f(x)<a}
846:there is a neighbourhood
134:
67:
46:
3335:I'm fine with not using
3226:{\displaystyle -\infty }
3174:{\displaystyle -\infty }
2226:https://mathoverflow.net
2136:{\displaystyle +\infty }
2100:there is a neighborhood
2023:{\displaystyle f(x): -->
1634:{\displaystyle (X,\mu )}
1570:18:04, 4 July 2012 (UTC)
1545:04:08, 28 May 2012 (UTC)
1237:00:25, 27 May 2005 (UTC)
1227:23:31, 26 May 2005 (UTC)
1134:14:58, 3 July 2011 (UTC)
141:project's priority scale
2495:I have now fixed this.
1578:closed convex functions
1552:closed convex functions
1529:closed convex functions
530:{\displaystyle x\neq 0}
98:WikiProject Mathematics
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1860:convergence in measure
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1177:. The definition for
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2910:Set-valued function
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2600:{\displaystyle x=0}
2464:exactly when it is
2373:Continuous function
2350:Sounds good to me.
1101:-\infty \}}" /: -->
594:{\displaystyle x=0}
215:) is a property of
3392:The-erinaceous-one
3365:The-erinaceous-one
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2736:The-erinaceous-one
2722:, or create a new
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2546:{\displaystyle f}
2113:{\displaystyle U}
2098:f(x_{0})}" /: -->
2030:-\alpha }" /: -->
1923:{\displaystyle f}
1832:{\displaystyle f}
1812:{\displaystyle f}
1560:comment added by
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1218:is important for
1035:-\infty \}}": -->
859:{\displaystyle U}
844:f(x_{0})}" /: -->
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1583:Sławomir Biały
1524:
1521:
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1504:
1500:
1496:
1439:
1436:
1435:
1434:
1420:
1419:
1418:
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1416:
1372:order topology
1367:
1366:Order topology
1364:
1363:
1362:
1312:
1311:
1284:
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1120:
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914:
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908:
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880:
876:
855:
833:
828:
824:
820:
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811:
788:
784:
763:
743:
723:
703:
683:
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665:
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637:
634:
631:
610:
590:
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584:
564:
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498:
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382:
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273:
269:
236:
213:semicontinuity
187:
183:
162:
159:
157:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
3432:
3421:
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3410:
3401:
3397:
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3387:
3382:
3381:
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3379:
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3366:
3362:
3354:
3350:
3346:
3334:
3333:
3332:
3328:
3324:
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3316:
3315:
3314:
3313:
3309:
3305:
3301:
3296:
3290:
3288:
3283:
3275:
3271:
3267:
3262:
3245:
3241:
3217:
3194:
3188:
3165:
3145:
3142:
3134:
3130:
3123:
3100:
3097:
3089:
3085:
3078:
3070:
3054:
3051:
3048:
3045:
3032:
3027:
3025:
3021:
3017:
3013:
3012:
3011:
3007:
3003:
2999:
2997:
2993:
2989:
2984:
2983:
2982:
2981:
2977:
2973:
2969:
2962:
2953:
2937:
2933:
2929:
2925:
2923:
2919:
2915:
2911:
2907:
2906:
2905:
2901:
2897:
2887:
2886:
2885:
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2859:
2840:
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2814:
2809:
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2799:
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2755:
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2725:
2721:
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2702:
2698:
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2690:
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2687:
2683:
2679:
2674:
2672:
2668:
2664:
2660:
2653:
2644:
2640:
2636:
2632:
2594:
2591:
2588:
2563:
2560:
2540:
2532:
2531:
2530:
2529:
2525:
2521:
2510:
2506:
2502:
2498:
2494:
2493:
2492:
2491:
2487:
2483:
2479:
2475:
2471:
2467:
2463:
2462:uniformizable
2454:
2438:
2434:
2430:
2426:
2422:
2418:
2414:
2410:
2409:
2408:
2404:
2400:
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2307:
2303:
2298:
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2292:
2288:
2284:
2278:
2273:
2272:
2269:
2265:
2261:
2257:
2253:
2252:
2251:
2250:
2246:
2242:
2235:
2232:
2231:
2230:
2227:
2223:
2218:
2216:
2207:
2199:
2195:
2191:
2187:
2186:
2185:
2181:
2177:
2173:
2172:
2171:
2168:
2162:
2161:
2160:
2159:
2155:
2151:
2147:
2127:
2107:
2080:
2076:
2069:
2066:
2063:
2054:
2049:
2047:
2043:
2035:
2033:
2017:
2014:
2011:
2005:
1999:
1978:
1975:
1969:
1963:
1960:
1940:
1937:
1917:
1904:
1902:
1901:
1897:
1893:
1888:
1885:
1879:
1875:
1871:
1867:
1861:
1857:
1856:
1855:
1854:
1850:
1846:
1842:
1826:
1806:
1796:
1788:
1786:
1785:Fatou's lemma
1764:
1761:
1755:
1729:
1726:
1723:
1715:
1711:
1690:
1667:
1664:
1661:
1653:
1649:
1625:
1622:
1619:
1607:
1606:
1605:
1604:
1603:
1597:
1593:
1589:
1585:
1579:
1575:
1574:
1573:
1571:
1567:
1563:
1559:
1553:
1547:
1546:
1542:
1538:
1534:
1530:
1522:
1518:
1514:
1510:
1494:
1490:
1489:
1488:
1485:
1481:
1477:
1473:
1469:
1461:
1459:
1455:
1451:
1450:92.141.19.110
1447:
1437:
1433:
1429:
1425:
1421:
1414:
1410:
1409:
1408:
1405:
1401:
1400:
1399:
1395:
1391:
1387:
1386:
1385:
1384:
1381:
1377:
1373:
1365:
1361:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1322:
1321:
1320:
1317:
1310:
1307:
1303:
1302:
1301:
1300:
1297:
1279:
1273:
1267:
1261:
1248:
1242:
1238:
1235:
1231:
1230:
1229:
1228:
1225:
1221:
1203:
1190:
1187:
1184:
1156:
1153:
1150:
1138:
1136:
1135:
1131:
1127:
1118:
1114:
1110:
1106:
1082:
1079:
1073:
1067:
1064:
1061:
1058:
1055:
1049:
1044:
1040:
1018:
1010:
1009:
1008:
1007:
1003:
999:
998:130.56.234.55
980:
972:
968:
961:
941:
938:
935:
915:
912:
906:
900:
878:
874:
853:
826:
822:
815:
812:
809:
801:if for every
786:
782:
761:
741:
721:
701:
678:
675:
672:
669:
663:
657:
635:
632:
629:
608:
588:
585:
582:
559:
556:
550:
544:
524:
521:
518:
496:
492:
487:
483:
480:
477:
471:
465:
439:
433:
419:
416:
404:
398:
394:
390:
386:
385:
384:
383:
380:
376:
372:
366:
365:
364:
363:
360:
358:
341:
337:
332:
328:
324:
320:
298:
294:
271:
267:
258:
254:
250:
234:
226:
222:
218:
217:extended real
214:
210:
206:
201:
185:
181:
172:
168:
160:
158:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
3360:
3359:
3291:
3281:
3280:
3158:also equals
2957:
2675:
2648:
2520:62.141.176.2
2514:
2458:
2300:take it out.
2238:
2219:
2211:
2050:
2042:WP:TECHNICAL
2039:
1908:
1889:
1886:
1883:
1841:sequentially
1840:
1797:
1793:
1608:
1601:
1562:137.82.36.82
1556:— Preceding
1548:
1526:
1462:
1441:
1412:
1375:
1369:
1352:
1348:
1344:
1340:
1336:
1332:
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1324:
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256:
252:
248:
212:
208:
202:
170:
166:
164:
156:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
2667:MOS:SEEALSO
1930:is lsc iff
1845:Marcosaedro
1466:—Preceding
1444:—Preceding
1095:-\infty \}}
259:at a point
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
3409:Categories
1404:Algebraist
1380:Algebraist
1357:Madmath789
893:such that
648:0}" /: -->
225:continuity
3386:PatrickR2
3370:PatrickR2
3323:PatrickR2
3304:PatrickR2
3287:WP:WEIGHT
3266:PatrickR2
3016:PatrickR2
2972:PatrickR2
2914:PatrickR2
2896:PatrickR2
2862:PatrickR2
2798:PatrickR2
2697:PatrickR2
2678:PatrickR2
2631:PatrickR2
2497:PatrickR2
2482:PatrickR2
2470:Hing Tong
2429:PatrickR2
2413:PatrickR2
2399:PatrickR2
2352:PatrickR2
2324:PatrickR2
2302:editeur24
2283:PatrickR2
2277:Editeur24
2260:PatrickR2
2241:PatrickR2
2190:editeur24
2176:PatrickR2
2150:PatrickR2
2092:f(x_{0})}
2024:-\alpha }
1424:PatrickR2
838:f(x_{0})}
458:given by
389:editeur24
371:editeur24
221:functions
3295:MOS:MATH
3116:. Then
3069:MOS:MATH
2968:MOS:MATH
2860:either.
1558:unsigned
1480:contribs
1472:Serban00
1468:unsigned
1446:unsigned
1126:Luzern66
928:for all
650:we have
625:0}": -->
219:-valued
3300:result.
2166:Mgkrupa
2046:Mgkrupa
1892:Aucross
1537:Zfeinst
1509:Mktyscn
1347:) : -->
1331:) : -->
1224:Clausen
1105:Zfeinst
139:on the
2394:(WPM).
1306:Deepak
1296:Deepak
36:scale.
2067:: -->
2012:: -->
1080:: -->
813:: -->
633:: -->
253:lower
249:upper
3396:talk
3374:talk
3349:talk
3327:talk
3308:talk
3270:talk
3020:talk
3006:talk
2992:talk
2976:talk
2932:talk
2918:talk
2900:talk
2880:talk
2866:talk
2802:talk
2740:talk
2701:talk
2682:talk
2635:talk
2524:talk
2501:talk
2486:talk
2433:talk
2417:talk
2403:talk
2381:talk
2356:talk
2342:talk
2328:talk
2306:talk
2287:talk
2264:talk
2245:talk
2224:and
2194:talk
2180:talk
2154:talk
1991:iff
1976:<
1896:talk
1870:talk
1849:talk
1609:Let
1588:talk
1566:talk
1541:talk
1513:talk
1476:talk
1454:talk
1428:talk
1394:talk
1222:. --
1130:talk
1109:talk
1002:talk
913:<
575:for
537:and
511:for
393:talk
375:talk
211:(or
3339:and
3067:.
2516:-->
1745:to
1374:on
866:of
247:is
203:In
131:Mid
3411::
3398:)
3376:)
3351:)
3329:)
3310:)
3297::
3272:)
3221:∞
3218:−
3169:∞
3166:−
3146:ε
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3101:−
3043:∀
3022:)
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2994:)
2978:)
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2920:)
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2777:⇉
2768:Γ
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2015:−
1979:α
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1277:⌈
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682:∞
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673:ϵ
642:0}
630:ϵ
563:∞
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522:≠
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42::
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