Talk:Continuous function

message: The image is reproduced here for clarity and completeness:
This image is referring to this definition of continuity at a point between two topological spaces:
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:
* , page 4 definition 4.6.
* , page 1 definition 1.1.
* , page 1 Theorem 18.1, 4th item.
* , first paragraph
Now, one could argue that they are self-published works, so here a (more difficult to find) published work mentioning this definition:
* 
Also U is not chosen after V, a U such that f(U) \subseteq V should just exists for (every) V (neighborhoods) and the image just show that by picturing one such U and illustrating what f(U) \subseteq V means . One could say that the picture doesn't shows every V (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. Pyrrhonist05 (talk) 18:29, 8 November 2017 (UTC)
thread_title:  Image in section "Continuous functions between topological spaces" is right 

message: The wiki claims that these are identical to their curvature continuity (Ck) counterparts, but they are not. The geometric continuity (Gk) 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. Boris Tsirelson (talk) 06:23, 19 January 2019 (UTC)
thread_title:  Incorrect definition of ''G''<sup>k</sup> continuity 

message: Shouldn't this be "uniformly continuous", and not just "continuous"? There is a discrepancy here between this page and the [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.  — Preceding unsigned comment added by 2603:9001:2902:1300:849B:38FF:720A:4408 (talk) 05:36, 24 September 2019 (UTC)
replies: text: There is no discrepancy, as, here, x_0 is defined before \delta, which implies that \delta may depend on x_0. D.Lazard (talk) 06:36, 24 September 2019 (UTC)
thread_title:  On the "Weierstrass and Jordan definition" 

message: 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! Houseofwealth (talk) 01:11, 16 January 2020 (UTC)
thread_title: Definition in terms of limits of functions

message: Greetings maintainers of this page. In the section "Definition in terms of limits of functions" you give the standard definition as 
\displaystyle \lim_{x \to c} f(x) = f(c). 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.
99.171.179.102 (talk) 21:14, 20 February 2021 (UTC)
thread_title: Error in limit definition

message: 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 x\mapsto \frac 1x 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.
--Felix Tritschler (talk) 11:50, 6 August 2022 (UTC)
replies: text: 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. 67.198.37.16 (talk) 06:31, 29 December 2023 (UTC)
replies: text: Please have a look at my reply to your other comment (IP address 67.198.37.16 from 29.Dec.2023). --Felix Tritschler (talk) 11:05, 12 April 2024 (UTC)
thread_title:  Suggestion to change the article based on the discussion 'Please remove the "highschool definition" ' 

message: 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. 120.51.141.38 (talk) 05:51, 17 April 2023 (UTC)
replies: text: As this section is a subsection of , 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. D.Lazard (talk) 09:01, 17 April 2023 (UTC)
thread_title:  Definition in terms of neighborhoods has issues