Is the support of the Gaussian finite or infinite? Considering that as $x \to \pm \infty$ ; $e^{-\frac{x^2}{2}} \to 0$, is the support finite or infinite? A simple enough question, but enough to make me scratch my head. I feel that it's almost a metaphysical question.  
What would it mean to have a probability distribution with infinite support? I know it's only a very good approximation, but for the theory to say that the support should be infinite doesn't sit well with me. On the other hand, intuitively, the closure of the support is $[-\infty,\infty]$ which implies that the support is infinite. Maybe it's a stupid question but it's bugging me a bit. Any and all thoughts are welcome!

Infinite. For every $x$, $e^{-x^2/2}>0$.
