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I've noticed that in analysis we often treat the unit-interval $[0,1]$ differently from $[0,\infty)$, particularly in improper-integration (but certainly not limited to).

By lieu of example, consider proving that the Gamma function converges; i.e., the integral exists. The Gamma function is defined as follows:
$$ \Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\,. $$

Typical proofs I've encountered consider the two cases over the interval $[0,1]$ and $[0,\infty)$. This is because of the properties of $t^{z-1}e^{-t}$ over $[0,\infty)$. (Sorry, I'm not going to go in detail here; it's just an example.)

However, this has me wondering: what makes the unit-interval $[0,1] \subset \mathbb R$ so special? Although I'm mindful that I may be splitting-hairs, I'm seeking to understand if there's some concept that generalizes the properties of the unit-interval; perhaps this suggests why we may often have to treat it differently in, for example, integration-problems? I'm looking for something related to the closed multiplicative (group under multiplication?) of $[0,1]$.

As a start, I know that from elementary calculus, $ \lim_{x -> \infty} a^x = 0 $ if $a \in [0,1)$, $a^x=1$ if $a=1$ and $a^x=\infty$ if a>1. I'm thinking my answer lies somewhere in field-theory / group theory under the operation multiplication. Obviously the question is open-ended, but I'm hoping there is some general property about the interval between the multiplicative and additive identities of $\mathbb R$ that perhaps explain why we may often have to treat it differently.

(Again, I apologize for being so vague; just looking for someone to direct me to further reading/subjects/theorems.)

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    Well, you may think as $[0,1]$ as the "generator" of any other finite interval, formally I'd think about it as $a\cdot[0,1]=[0,a]$. Similarily, you can think about $[a,b]=b\cdot[0,1]-a\cdot[0,1]$ so usually when one proves some important properties of a function or expression over $[0,1]$ it is usually the case we can extend its properties by a "dilatation" of the unit interval. I can give you an example precisely for the $\Gamma$ function, if you want to.2012-04-18
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    Another thing to note is that (as stated above) $[0,1]$ essentially generates all *compact* intervals, i.e., those closed intervals that have finite endpoints. On the other hand $[0,\infty)$ is a simple non-compact closed interval, since it is not bounded. It can also be used as generator for the closed non-compact intervals in a similar way.2012-04-18

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