Can the range of a variable be inclusive infinity? 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 &lt; x &lt; 1 $. My question is whether infinity can be written as inclusive when stating the range.
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
$$f(x)=\begin{cases}x &amp; \mathrm{if } \quad x\neq 0\\\infty &amp; \mathrm{if }\quad x=0\end{cases}$$
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.