# Definition talk:Piecewise Continuous Function

The link to the work cited is to the 4th edition. Can someone do some research to establish the chronology of the actual publication dates, in particular the date of its its original publication? --prime mover (talk) 15:42, 18 July 2013 (UTC)

- I have done some google searching but have been able to find only the third edition:
- Tyn Myint-U and Lokenath Debnath: Linear Partial Differential Equations for Scientists and Engineers; (3rd ed.) Springer, New York, NY (2006) Ivar Sand (talk) 08:36, 22 July 2013 (UTC)

Explanation for using "partition":

- I have checked the references below and found that 2 of 4 use "partition" and the others just use "points".
- I have also checked the references in the talk page of Piecewise Continuously Differentiable Function and found that 3 of 10 use "partition". Ivar Sand (talk) 07:48, 22 August 2013 (UTC)

Is it an idea to move the subsection "Possible properties of piecewise continuous functions" to here (the talk page)? It just contains a list of possible things to do. I do not know whether the work in that subsection will be done or how long it will take; personally, I would like to do only a fraction of it. Ivar Sand (talk) 09:09, 22 August 2013 (UTC)

- It's a placeholder so as to indicate that it is work that needs to be done.

- In ProofWiki, every single statement of mathematics (no matter how trivially it follows from a definition) needs to be expressed on its own separate page, complete with a proof (even if that proof is only of one line).

- Hence each of the statements made within this section (in spite of the "it seems obvious" remark) needs to be justified with a rigorous proof.

- ProofWiki, you will note, has a house style which is considerably more rigid than that of many similar wikis. There are many conversations on discussion pages which give an indication as to how this style evolved. In this context, we try to avoid naming sections "Comments" and so on: if a statement is required as part of the definition, then it will be so included. If a statement is *not* so required, then it is moved to a separate page accessible via a link in the "Also see" section, which indicates that it contains information peripheral to the statement, and a fuller account of it can be found in the page behind that link.

- If you do not wish to undertake the work as described above, then you do not have to. The plan is that it will be achieved in due course by someone who wishes to take on the effort of completing all incomplete pages (which can be found by following the links at the left hand side of this page). --prime mover (talk) 11:39, 22 August 2013 (UTC)

I wonder if the refactor template in the "Also defined as" section could be removed. This is because the other definitions in the "Also defined as" section were found outside of ProofWiki, and a search for "Piecewise" in ProofWiki reveals that they are not mentioned in ProofWiki. Therefore, the separate pages with /Variant 1, /Variant 2 etc. mentioned in the refactor template should not be needed. Is this correctly understood? Ivar Sand (talk) 08:38, 1 November 2013 (UTC)

- No, please leave them. They are reminders to us that we intend to do some work in this area in due course. --prime mover (talk) 11:04, 1 November 2013 (UTC)

A statement on combinations of piecewise continuous functions is the only statement left in the Comments section now.
I doubt whether combinations of piecewise continuous functions constitute an area that is worth exploring because I guess that such combinations are not common and maybe even nonexistent.
(However, piecewise continuous functions multiplied by continuous functions exist.) Ivar Sand (talk) 08:37, 8 November 2013 (UTC)

- Every statement made needs to be backed up by a page proving that statement. In particular, "it seems obvious" and its ne'er-do-well brothers "clearly" and so on are no acceptable on ProofWiki. This is why the maintenance label exists. It is there to remind us that there is still work to be done.

- In particular, the wording of this section expresses a belief that the truth of the statement being made is uncertain: "possibly" and "seem" give the game away. Until this is established by explicit demonstration the maintenance tag remains. --prime mover (talk) 11:22, 8 November 2013 (UTC)

- I understand. But how about moving the contents of the Comments section here? I think here is where the current comment in the Comments section should have been put in the first place. And then the Comments section, which is a non-standard section anyway, could be removed from the definition page. In this way, the information in the Comments section would still exist (but elsewhere) in ProofWiki while the readers of the definition would not be bothered with information that, as I guessed in my previous post, may have a doubtful value. Ivar Sand (talk) 09:14, 12 November 2013 (UTC)

- No, leave as it is, then it acts as a reminder for work to be done. That's how it works here. --prime mover (talk) 12:14, 12 November 2013 (UTC)

## Contents

## Other definitions of piecewise continuous

1. I have searched the list at http://www.proofwiki.org/wiki/ProofWiki:Community_Portal#Magazines for other definitions of "piecewise continuous" and found:

Agarwal and O’Regan:
$f$ needs not be defined at $x_i, i$∈{0,…,n}.^{[1]} ^{[2]} (The search function of maa.org was used.)

2. I have searched the list at http://www.proofwiki.org/wiki/ProofWiki:Community_Portal#Wikis_and_Encyclopedias for other definitions of "piecewise continuous" and found:

- mathworld.wolfram.com:
resorts from being as specific as in (2) and says instead "certain matching conditions are sometimes required".^{[3]}

- planetmath.org:
lacks (2).^{[4]}

3. I have found these on the Internet (I have done only a limited search):
- Advanced Calculus: MATH 410 Notes on Integrals and Integrability, Professor David Levermore:
(2) is replaced by the requirement that $f$ be bounded.^{[5]}
Ivar Sand (talk) 10:05, 24 July 2013 (UTC) and 9 August 2013 (UTC)

## Are definitions equivalent?

Are these definitions of Piecewise Continuous equivalent? I would work it out, but I'm busy on something else. --prime mover (talk) 19:29, 4 March 2015 (UTC)

- It would seem not. Consider $f(x) = \sin \frac1x$ with $(0,1)$ as one of the intervals. It fits definition 2, but not 1. — Lord_Farin (talk) 20:26, 4 March 2015 (UTC)

- In that case we need a different design paradigm here than "/Definition 1" and "/Definition 2" -- not sure what, but what is being used here is for two definitions which are equivalent. Would it be better to implement these as "Variant 1" and "Variant 2"? Can we unearth alternative terms for either of these "piecewise continuous" definitions which differentiates them from each other? I am worried that the differences are going to become opaque, and the wrong one will be used in crucial places, thereby compromising the integrity of the proofs that invoke them. --prime mover (talk) 20:49, 4 March 2015 (UTC)

- I would prefer using distinct terms if they exist; the situation that this is not the case has until now been resolved by coining terms. I can't think of something right now. We will have to think about a good approach. I'd also like to hear what Ivar has to say on this. — Lord_Farin (talk) 21:12, 4 March 2015 (UTC)

- I find /Variant 1, /Variant 2, which prime mover suggested originally (sorry, prime mover!), a bit better than /Definition 1, /Definition 2 since we already know that we are talking about definitions, also, they suggest that the definitions may be overlapping.
- Maybe /Variant Limit, /Variant Bounded or /Limit Variant, /Bounded Variant would be better than /Variant 1, /Variant 2.
- Instead of "piecewise continuous", "piecewise continuous (v. Limit)" and "piecewise continuous (v. Bounded)", where "v." stands for "variant", might be used for definitions 1 and 2 respectively, but I don't know whether this is usual in English.
- Now, definition 1 implies definition 2, so definition 1 is subordinate to definition 2. Maybe this suggests that "piecewise continuous" should be used for definition 2 and, perhaps, "piecewise continuous (v. Limit)" for definition 1.—Ivar Sand (talk) 15:17, 5 March 2015 (UTC)

- I'm against anything like "v. Limit" because it is not very descriptive. I would indeed be inclined to define PC as definition 2, and refer to definition 1 as "PC with one-sided limits" or something similar. — Lord_Farin (talk) 18:12, 5 March 2015 (UTC)

- LF's suggestion works for me.

- I plan on working on these pages myself in due course, but at the moment I am disciplining myself to finishing off Euclid, which is painful and tedious. So I'm not starting on another project till I finish that. But I'm half way through oook XII now, so not much further to go. --prime mover (talk) 20:02, 5 March 2015 (UTC)

- Note that my (limited) experience with google searching for piecewise continuous functions indicates that Definition 1 is quite common whereas Definition 2 seems rare. Also, Definition 1 is linked to Definition 1 of piecewise continuously differentiable functions.--Ivar Sand (talk) 10:06, 18 March 2015 (UTC)

- 1. If, generally speaking, two definitions, say "/Definition 1" and "/Definition 2" are equivalent, maybe "/Formulation 2" would be better than "/Definition 2" and maybe also better than "/Variant 2". (This is not relevant here so far, but it is relevant to piecewise continuously differentiable functions.)

- 2. Also, the term "Variant", or some synonym for it, may have a potential use on the current definition page. I am thinking of the first item of the Also defined as section: "f need not be defined at the points $x_i$". This seems to me not to qualify as a future, independent definition but rather a variant of either of the two definitions that have been entered so far. It may give rise to "/Definition 1/Variant 1" and "/Definition 2/Variant 1". I would expect these variations to have the same theorems as respectively /Definition 1 and /Definition 2 (but with different proofs).

- Something similar goes for the second item of the Also defined as section: "The subdivision above can be infinite when the domain of f is unbounded". This would not qualify as a future definition either but would give rise to variations not only to the existing definitions but to the variations of them as well, like this: "/Definition 1/Variant 2", "/Definition 2/Variant 2", "/Definition 1/Variant 1/Variant 1", and "/Definition 2/Variant 1/Variant 1".

- Maybe it would be a better idea, for the two first items of the Also defined as section, to make two new paradigms that are variants of the current paradigm, like this: "/Variant 1/Definition 1" and "/Variant 1/Definition 2", and "/Variant 2/Definition 1" and "/Variant 2/Definition 2".

- Currently concentrating on finishing off Euclid. I will attend to this matter as and when it reaches the top of my schedule. --prime mover (talk) 09:56, 16 March 2015 (UTC)

- I'm sceptical about changing the way we handle multiple definitions of the same and/or closely related concepts, even if it would be determined some other way would theoretically constitute an improvement. You do realise that this change would either introduce inconsistency or require an absolutely ludicrous amount of work? — Lord_Farin (talk) 22:56, 16 March 2015 (UTC)

## Two more definitions

I propose another two definitions of piecewise continuous functions in addition to the two existing ones.

In my opinion, as I pointed out above, the definitions indicated in the Also defined as section are more like variations of existing definitions, or variations of the existing paradigm of definitions, than independent definitions, so they would not qualify.

The current paradigm has a certain structure consisting of two parts: (a) A theorem for Definition 1 is transformed into a part of Definition 2, and that is the theorem for boundedness; also, (b) Definition 1 implies Definition 2. This structure should fit the next definitions as well.

According to this structure, Definition 2 has a theorem for Riemann integrability that could correspond to a requirement of Riemann integrability in a new Definition 3. This requirement of Riemann integrability should allow for improper integrals because otherwise Definition 3 would be equivalent to Definition 2, I believe. By google searching I have found such a definition for piecewise continuity.^{[6]} (This reference talks about $\R^3$, but it's the idea that counts.)

The next logical step, Definition 4, seems to me to be a definition that is mentioned on this page, but that for some reason did not survive to the Also defined as section on the definition page, and that definition is the same as Definition 1 and 2 except that it lacks (2).^{[4]}

These four definitions (should) fit the current paradigm structure, i.e. they satisfy requirements (a) and (b).

I plan to upload these two definitions tomorrow.--Ivar Sand (talk) 10:15, 18 March 2015 (UTC)

- I have a problem with this. --prime mover (talk) 20:11, 18 March 2015 (UTC)

- I'm not sure I am able to explain exactly
**why**I have a problem with this except: a) it's not the way it's been done in the past (that is, to hunt down every possible definition that can be found on the web and post it up) and b) introduce each different definition as equivalent in nature to each other -- when in this case they are clearly not.

- The only other instances of different definitions on this site have been handled by proving their equivalence, or by deep analysis of the works which use a different definition and sideline the different approaches in separate threads. The only example of this I can find is the difference between the formulation of $1$-based and $0$-based natural numbers, and even then I don't think it's analogous.

- My view is that "Definition 1": that is a function which, on each closed interval of the partitioning of the interval, is continuous at either end. I believe that the "Definition 2" is not
**really**"piecewise continuous, unless it can be demonstrated to be the same as Definition 1 (which I know it has not been but I am looking at the Continuity Property and think there's an implicit equivalence here that can be exploited).

- But feel free to post up your further definitions. Would be good to see sources cited, of course. I will approach this section later (when I want to; I take things at my own pace) and restructure it into something that approaches the house style. --prime mover (talk) 22:32, 18 March 2015 (UTC)

- My idea behind searching the web is not to hunt down every definition, but to make the set of definitions here in a sense complete.

- When I first uploaded a definition of piecewise continuous functions I did so in the belief that there was only one, and I felt that I would like to repair that error. That's why I started to search for other definitions. But on the way I saw a pattern of definitions emerge. The two existing definitions and the two new ones fit nicely together. They are overlapping but not equivalent. The fourth one is the most general one having the weakest requirements. A user can pick out the definition that suits his or her purposes best.

- I have found all four definitions on the web, but I had a problem finding the third one. I felt it was important to find that definition in use as well because otherwise it would be an invention, and that might not be right to present to the reader. This is because I expect that a reader would look for something in use rather than a definition that is made up just to fit some theoretical scheme that the reader might find uninteresting or speculative. I am thinking from my background as a physicist here: the truth does not lie in theory alone, one must determine that the phenomenon in question exists in reality as well. You mathematicians think maybe that a true statement is interesting in its own right.

- I don't quite understand what you are thinking about concerning the definitions 1 and 2. As you can see from the theorem in the Relations between Definitions section on the definition page, they are not equivalent. The idea behind them is that Definition 2 is more general than Definition 1 in that the limits in (2) in Definition 1 need not exist. According to Definition 2 $f$ is allowed to swing freely up and down so long as it does so within bounds. By the way, notice that the individual definition pages have listed on them theorems that do not appear on the multi-definition page.--Ivar Sand (talk) 16:15, 19 March 2015 (UTC)

- I'll let you get on with it then. All yours. I will see how it looks when you've gone as far with it as you feel you can, and we will then be able to review what may need to be done to fit it all better into the house style structure. --prime mover (talk) 21:51, 19 March 2015 (UTC)

## New name proposals

Now that the definitions themselves seem to have settled, I propose the following naming convention:

- Piecewise Continuous Function with One-Sided Limits
- Bounded Piecewise Continuous Function
- Integrable Piecewise Continuous Function
- Piecewise Continuous Function

If this is agreed upon, work can be started on rationalising the various page names. — Lord_Farin (talk) 14:53, 16 May 2015 (UTC)

- Since "piecewise continuous function" is a generic term for all definitions, I think that in order to distinguish the fourth definition from this generic term, it should have some additional text too, like the other three definitions have. I am thinking of texts like "basic", "unconditioned", "unqualified".--Ivar Sand (talk) 07:43, 26 May 2015 (UTC)

- The way I see it, since the other three encompass the fourth, this convention makes sense. The only context in which I have encountered "basic" things is in the context of Definition:Basic Open Set, where it is not the case that every open set is a
*basic*open set. Given the thorough investigation and the formulation of the fourth definition, I would consider it justified to have this as the "basic", "unqualified" PC function. The adjectives "unconditioned" and "unqualified", however, would otherwise be antinomous because they*do*qualify the kind of PC function under consideration.

- The way I see it, since the other three encompass the fourth, this convention makes sense. The only context in which I have encountered "basic" things is in the context of Definition:Basic Open Set, where it is not the case that every open set is a

- That all notwithstanding I agree it is very important to have a clear "Also defined as" section, and possibly other mechanisms, in place to make sure that people writing pages actually use the correct definition. This however can never serve as a substitute for reading through the work being posted, so we'll have to be vigilant in any case. Naturally I'm curious to your arguments as to why it should be distinguished, I just set out my point of view. — Lord_Farin (talk) 14:55, 26 May 2015 (UTC)

- What I mean with "to distinguish" is "to separate"; I don't mean that definition 4 should be distinguished. Since I am not English-speaking, some of my English texts are unclear, and I am sorry about the problems that this is bound to cause from time to time.

- Perhaps it is an idea to test your naming on a few pages that talk about piecewise continuity and, in particular, definition 4, for instance Piecewise Continuous Function does not necessarily have Improper Integrals. That page would or might get the name: "Integrable Piecewise Continuous Function implies Piecewise Continuous Function". I fear that this name might confuse the readers. --Ivar Sand (talk) 08:18, 29 May 2015 (UTC)

- I wouldn't keep that proof up. It's pointless if we define an "integrable PC function is a PC function that is integrable" with appropriate links. And, instead of having "Integrable PC Function is PC Function with One-Sided Limits" we would say "Integrable PC Function has One-Sided Limits". Does that make sense to you? — Lord_Farin (talk) 09:55, 30 May 2015 (UTC)

## House style

@Ivar: The house style tidy templates are there because of the "Theorems" subsections, which are normally placed in the "Also see" section. But before embarking on that work, please consider the name changes suggested above. Implementing those is also necessary to bring the page names in line with house style, and is more important. — Lord_Farin (talk) 17:18, 19 May 2015 (UTC)

- Right now I am busy working on correcting work that I have done previously. This work is not meant to be in conflict with your page name suggestions although it may seem to be.

- Thanks for the note about the "Theorems" subsections. When a tidy template is put on one of "my" pages, I always wonder what is wrong because the text associated with it is not specific.--Ivar Sand (talk) 10:39, 20 May 2015 (UTC)

- The rename suggestions are a fine intermediary step, but I wouldn't go as far as executing them, only to repeat the exercise after the final names have settled down. As to your remark on the tidy templates, thanks for that. I'll try to include some more specific pointers and links in those in the future (although these shouldn't be considered exhaustive, but do drop me a note if you think a page's state does no longer warrant a tidy). — Lord_Farin (talk) 16:22, 20 May 2015 (UTC)

## Refactoring

The first stage of refactoring is complete. --prime mover (talk) 19:38, 19 March 2018 (EDT)

- Thanks for your good and extensive work, prime mover! I would have preferred to keep Definition 4 myself, but it also works the way you have done it. --Ivar Sand (talk) 04:35, 21 March 2018 (EDT)

- Well, we have kept Definition 4, but not as a separate definition, but instead as the main and most basic (and weakest) definition. The other definitions are more restrictive, and impose further conditions on top of that -- hence the need to put them in separate pages. (We might want to decide whether to reorder the remaining definitions from weaker to stronger, but as they are now no longer numbered but
*described*, this is now a trivial exercise that requires no further renaming.) But it is of little benefit to implement a separate page for "just" continuous on the subintervals.

- Well, we have kept Definition 4, but not as a separate definition, but instead as the main and most basic (and weakest) definition. The other definitions are more restrictive, and impose further conditions on top of that -- hence the need to put them in separate pages. (We might want to decide whether to reorder the remaining definitions from weaker to stronger, but as they are now no longer numbered but

- Clearly, to be any practical use at all, we
*do*need to further qualify the definition for specific purposes (like, for example, the underlying structure of a function must also include "has one-sided limits" before you can get its Fourier series, which is where I'm at right now), but in that case we*specifically*invoke that version of the definition at the point we need to.

- Clearly, to be any practical use at all, we

- We also need to revisit whether the "bounded PCF" can indeed always be relied upon to be Riemann integrable, as I have reason to believe this might not be the case. --prime mover (talk) 10:10, 21 March 2018 (EDT)

- I still don't understand: what would be the purpose of keeping Definition 4 page? I need guidance here, as I am clueless. --prime mover (talk) 05:44, 23 March 2018 (EDT)

- A Definition 4 page (with a new name) would be used to link to in order to tell exactly what definition is meant, just as for the other three definitions. Today, the Definition:Piecewise Continuous Function page would be used for this purpose. However, this page talks about other definitions as well, which might confuse some readers. On the other hand, if one links to this page from some other page with the text "piecewise continuous" it should be clear for most readers that the definition meant is the first one on the Definition:Piecewise Continuous Function page as this definition is identified by "piecewise continuous". --Ivar Sand (talk) 10:11, 23 March 2018 (EDT)

- Sorry, still can't work out how it would be done. I can't think what it would be called, and I don't see how the page structure would be improved. --prime mover (talk) 14:09, 23 March 2018 (EDT)

## References

- ↑ Ravi P. Agarwal, Donal O’Regan: Ordinary and Partial Differential Equations (2009): §11: Definition 11.1
- ↑ Scroll down to "Piecewise Continuous and Periodic Functions" and press "Look Inside"
- ↑ http://mathworld.wolfram.com/PiecewiseContinuous.html
- ↑
^{4.0}^{4.1}http://planetmath.org/piecewise - ↑ http://www.terpconnect.umd.edu/~lvrmr/2004-2005-F/Classes/Integral.pdf
- ↑ A. G. Ramm and A. Rona, "Many-body wave scattering by small bodies and applications II", equation (10).