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)