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I've noticed some authors give the definition for Riemann integration in terms of a function $f$ with domain $[a,b]$. A definition might read "A function $f:[a,b]\to\mathbb{R}$ is Riemann integrable if ...". But don't we just want $f$ to be defined on $[a,b]$ (that is, for $[a,b]$ to be contained in the domain of $f$)? Why are we restricting $f$ in such a way? What's the point?

isthisreallife
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    Does "a function $f:U\to \Bbb R$ where $[a,b]\subseteq U$" really add anything to the definition? We can always restrict a function to some subset of its domain if we need. –  Dec 01 '16 at 20:11
  • Well, if a function $f$ is defined somewhere else than on $[a,b]$ then this definition applies to its restriction to $[a,b]$. Hence you would not gain any generality by just supposing that $[a,b]$ is contained in the domain of $f$. – NeedForHelp Dec 01 '16 at 20:12
  • When you restrict a function's domain, you're no longer talking about the same function. So now, we can't really phrase our proofs in a general way. We can't say something like "the square root function is integrable on any nonnegative interval $[a,b]$" because the square root function does not have domain $[a,b]$. The square root function is not of the form $f:[a,b]\to\mathbb{R}$. – isthisreallife Dec 01 '16 at 20:17
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    There is a simple and unambiguous geometric meaning to $\int_a^b$. The meaning of $\int_A$ for an arbitrary set $A$ isn't well-defined – user251257 Dec 01 '16 at 20:21
  • user251257: I'm not sure what relevance your comment actually has. A definition can be stated as "A function $f$ is Riemann integrable on $[a,b]$ if ...". Many authors do this as well. It's not ambiguous if the function is defined on a larger set of real numbers. – isthisreallife Dec 01 '16 at 20:22
  • They are just stating that the Riemann integral is only meaningfully defined on some set $[a,b]$, which seems to be what you're getting caught up on. – Logan Clark Dec 01 '16 at 20:24
  • I updated my comment. It's not ambiguous if the function has an arbitrary $D\subseteq\mathbb{R}$ domain because we write "$f:D\to\mathbb{R}$ is Riemann integrable on $[a,b]$". We're still talking about integrability on a closed bounded interval. – isthisreallife Dec 01 '16 at 20:28
  • @isthisreallife not every set admit an approximation by compact intervals. – user251257 Dec 01 '16 at 20:35
  • @isthisreallife why make life complicated if you have no benefits from it? – user251257 Dec 01 '16 at 20:36
  • I don't think I said that an arbitrary set can be approximated by compact intervals (what does that even mean?). I said that the property "$f$ is Riemann integrable on $[a,b]$" can still hold (it can be meaningful) even if the domain of $f$ isn't exactly the set $[a,b]$. I gave you a benefit in the third comment to this conversation. – isthisreallife Dec 01 '16 at 20:41
  • @isthisreallife you assume the notion is more general which it isn't. Everything in your definition depends on $f$ on $[a,b]$ only. So what do you need $D$ for if you can't define the integral on it? – user251257 Dec 01 '16 at 20:50
  • Ok, I retract saying it is more general. For me, it is more aesthetically pleasing to have a certain number of well known objects (such as the square root function) and to state results in terms of those well known objects (instead of constructing more and more objects). I believe we should be saying "the square function has such and such properties" instead of saying "every restriction of the square function to a compact interval has this property". – isthisreallife Dec 01 '16 at 21:02
  • Also when we allow an arbitrary domain, the definition becomes more permissive. To me, it seems that it has a wider applicability. If we're handed a function, it's more often the case that the function has an arbitrary (interval) domain. Forming a new function every single time seems unnecessary. – isthisreallife Dec 01 '16 at 22:04
  • @isthisreallife in your definition one always need to say $f$ is Riemann integrable on $[a,b]$. I think it's rather cumbersome. When it comes personal preference, I can't and don't want to persuade you of anything. – user251257 Dec 02 '16 at 01:16

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