Let me rephrase my question.

Bolzano's Thm states:

Let f(x) be continuous on [a,b].

f(x) changes sign on [a,b] => f(x) has a zero on [a,b]

I have a problem where f(x) changes sign but on the interval of all real numbers, from negative infinity to infinity. I would like to use Bolzano's Theorem, but my interval is open, not closed as worded in the theorem. Any suggestions?