Talk:Existence of Integral on Union of Adjacent Intervals

Regarding the presentation, I would prefer splitting the statement "$\int = \int + \int$, provided both exist" and the statement "$\int$ exists iff $\int + \int$ exists". I'm not sure whether these should be on the same page, but I'd say they do (perhaps using subpages). This type of proof abounds in measure theory and real analysis, so either we already thought of some good way to present this, or we should do this now so that we can mimic the approach for future applications. &mdash; Lord_Farin (talk) 09:46, 27 April 2015 (UTC)


 * I don't know if the following solves the problem, but I think it is a (small) improvement:




 * Then:
 * if $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$, $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$ and $\left[{c \,.\,.\, b}\right]$
 * and
 * If $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$ and $\left[{c \,.\,.\, b}\right]$, $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$
 * and in either case:
 * $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$ --Ivar Sand (talk) 07:23, 28 April 2015 (UTC)


 * The usual way we state such a theorem as this is:


 * "$f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$ $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$ and $\left[{c \,.\,.\, b}\right]$."


 * Then the proof is split into two parts: "Necessary Condition" and "Sufficient Condition". Whether those two separate parts are put in separate pages and then each transcluded depends on how complicated the proof of each part is. --prime mover (talk) 10:54, 28 April 2015 (UTC)


 * ... note we already have a page stating $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$ so that does not need to be done here. In any case, it's a separate result and so belongs on a different page, by the nature of the structure of itself. --prime mover (talk) 10:56, 28 April 2015 (UTC)


 * I am aware of that page (I assume you are talking about Sum of Integrals on Adjacent Intervals), but that theorem does not fit because it requires the function to be continuous. --Ivar Sand (talk) 10:53, 29 April 2015 (UTC)


 * What I have gathered so far concerning what should (or could) be done:
 * - Write a new theorem "$f$ is Riemann integrable on [a..b] if and only if $f$ is Riemann integrable on [a..c] and [c..b].", and I propose the following name for it: Existence of Integral on Union of Adjacent Intervals. Its proof is based on the proof of the current theorem.
 * - The current theorem is changed to:
 * Let $f$ be a real function which is Riemann integrable on a closed interval $\left[{a \,.\,.\, b}\right]$, $a < b$.
 * Let $c$ be a point in $\left({a \,.\,.\, b}\right)$.
 * Then $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$
 * - The proof of the current theorem is simplified by using references to the new theorem. --Ivar Sand (talk) 11:00, 29 April 2015 (UTC)


 * As I said, the page proving $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$ already exists: Sum of Integrals on Adjacent Intervals. --prime mover (talk) 11:13, 29 April 2015 (UTC)


 * Your approaches can easily be unified by weakening the condition of Sum of Integrals on Adjacent Intervals, as the note there already suggests. &mdash; Lord_Farin (talk) 16:22, 29 April 2015 (UTC)


 * I think I could change the page Sum of Integrals on Adjacent Intervals by replacing the condition of continuity by a condition of integrability. However, I have a problem with its Sources section because here locally there is no library that contains the books I would need ... --Ivar Sand (talk) 11:32, 4 May 2015 (UTC)


 * My suggestion is to write a page with integrability instead of continuity. The existing page can stand, as a statement of the theorem for continuous functions, and it will have 2 proofs: Proof 1, which will say "as a continuous function is an integrable function, then ... (link to integrable function proof), and Proof 2, which is the existing poof. --prime mover (talk) 11:53, 4 May 2015 (UTC)


 * Separating the two seems natural, because continuity makes stuff a lot easier, allowing for more accessible proofs. On the other hand, we do want to document the most general result out there. &mdash; Lord_Farin (talk) 16:22, 4 May 2015 (UTC)