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)

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}. (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".

- planetmath.org: lacks (2).

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. Ivar Sand (talk) 10:05, 24 July 2013 (UTC) and 9 August 2013 (UTC)