Talk:Piecewise Continuous Function with One-Sided Limits is Darboux Integrable

Is "Additivity with respect to Interval of Integration" perhaps a reference to Sum of Integrals on Adjacent Intervals for Integrable Functions? &mdash; Lord_Farin (talk) 08:52, 27 September 2013 (UTC)


 * No. "Additivity with respect to Interval of Integration" presupposes that two of the three integrals exist whereas "Sum of Integrals on Adjacent Intervals" presupposes that the common integrand is continuous. Ivar Sand (talk) 09:17, 27 September 2013 (UTC)


 * The hypotheses may not match, but the proof certainly follows through for any $f$ satisfying your hypotheses. &mdash; Lord_Farin (talk) 09:21, 27 September 2013 (UTC)


 * It seems to me that in the proof of "Sum of Integrals on Adjacent Intervals" one needs the existence of the integral from a to b, and this is assured by the continuity of $f$. In our case, we need to prove that the integral from a to b exists. Ivar Sand (talk) 09:52, 27 September 2013 (UTC)


 * There is indeed some work to be done. I'll put up a call to Template:Expand. &mdash; Lord_Farin (talk) 09:58, 27 September 2013 (UTC)

I intended "Integrable" and not "integrable" in the title so this is an error. Sorry! Please correct the error. Ivar Sand (talk) 09:54, 27 September 2013 (UTC)


 * Done. &mdash; Lord_Farin (talk) 09:58, 27 September 2013 (UTC)

Does the suggestion "The result referenced from Apostol is to be extracted and put into in its own right" mean that Apostol's book is not copyright protected any more?--Ivar Sand (talk) 08:59, 10 March 2015 (UTC)


 * It means that all results referenced on must themselves be results on {{ProofWiki}. It is doubtful that a proof at this level is genuinely the intellectual property of Apostol (I am sure that these results have been kicking around since a lot longer than from 1967), so the construction of the proof cannot be considered as subject to copyright laws. --prime mover (talk) 10:45, 10 March 2015 (UTC)


 * I think the idea of a proof is not copyrighted, but the formulation of it is. Therefore, to enter the theorem and proof word by word into proofwiki from Apostol's book would be breach of copyright. So, the proof has to be reinvented to be included in proofwiki.--Ivar Sand (talk) 09:34, 11 March 2015 (UTC)


 * There was never an instruction, implicit or otherwise, to report a proof word for word. The point is, though, that in order to link to a proof, it is pointless to include a reference to a proof in a book which is not on line. You might want to familiarise yourself with the contents, structure and philosophy of this site. Every statement in every page links back to another statement or proof within.


 * The page herein references a proof: "Additivity with Respect to the Interval of Integration" for which there is no proof yet on . All there is is a link to a reference in a hard-copy. The contents of this proof need to be added to . We do not expect to copy Apostol word for word -- but the sense of the proof is needed. --prime mover (talk) 13:50, 11 March 2015 (UTC)


 * I think I understand the problem now, and I am sorry for having caused it. I'll try to find a solution that is in accordance with rules.--Ivar Sand (talk) 08:41, 13 March 2015 (UTC)

Can't call it that
The / in the name would split it up into two slash-separated halves.

My initial, medial and final thoughts on this is that we should never refer to these definitions as /Definition 1, /Definition 2, etc. but find a different way of defining them. The /Definition 1 technique is being used for equivalent definitions, and to diverge from that standard in this instance is courting confusion.

I have not spent any time thinking about it because this entire section is so opaquely written that my eyes glaze over when I even think about it.

Perhaps "Definition:Piecewise Continuous Function by One-Sided Limits", "Definition:Piecewise Continuous Function by Boundedness", "Definition:Piecewise Continuous Function by Improper Integrals" and "Definition:Piecewise Continuous Function by Continuity" would be better names for the actual definitions themselves, and then the specific differences between the four definitions can be brought into sharper focus. In particular, the equivalence proofs will then be easier to name. --prime mover (talk) 16:19, 25 October 2017 (EDT)