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 < x < 1 $. My question is whether infinity can be written as inclusive when stating the range.


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.
