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Can a range be $[0, \infty]$ or must it be $[0, \infty)$ because you can never quite reach infinity?

Clarification: $[0, 1]$ means $0 \leqslant x \leqslant 1 $, while $(0, 1)$ means $0 < x < 1 $. My question is whether infinity can be written as inclusive when stating the range.

Amin Shah Gilani
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4 Answers4

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By convention you use $(-\infty$ or $\infty)$, that is how you will always find it in the literature.

vounoo
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  • Yes, this was my question. Could you cite the fact that it's conventionally written with parenthesis and not brackets? – Amin Shah Gilani May 03 '15 at 21:56
  • well no, but if you think about it, the attribute of ) that makes it differ from ] is already encoded in $\infty$. Infinity is not a point to be reached. Conventionally we use (). – vounoo May 03 '15 at 22:12
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    May I suggest something? Try to phrase why you would want to write infinity]. This might help you, – vounoo May 03 '15 at 22:30
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Generally if including positive and negative infinities, one says that one is working with the "extended real line", usually denoted $\mathbb{R}\cup\{\infty,-\infty\}$. It's quite possible, consider the function $$f(x)=\begin{cases}x & \mathrm{if } \quad x\neq 0\\\infty & \mathrm{if }\quad x=0\end{cases}$$

Dan
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An example of where this is actually used are measure functions. For instance, the Lebesgue measure, $\lambda$ on (certain subsets of) $\Bbb R$ has the property that $$\lambda((a,b)) = b-a $$ when $a,b$ are finite, while, for example $$\lambda(\Bbb R) = \infty$$.

GPerez
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Can it be $e^\pi$? $\infty$ and $e^\pi$ are just two entities that you may or may not consider part of a set. You can perfectly include them and define functions on that set or taking values of that set.