Is it true that any nonzero function $f: \mathbb R \to \mathbb R$ which is either:
1) constant, or
2) a polynomial, or
3) $\exp$, or
4) $\log$, or
5) any finite combination of the above using addition, subtraction, multiplication, division and composition, (and individually considered as functions from $\mathbb R$ to $\mathbb R$),
Has a finite number of zeros?
I'm actually interested in asymptotic behavior of functions, so a function f(x), as far as I'm concerned, is any expression constructed using the syntax below, such that starting from some point c>0, f(x) is defined for all x>c and takes real values everywhere in this range. A function has the following syntax:
F --> real number F --> exp F --> ln F --> -F F --> F + F F --> F(F)
Hope this specifies exactly what I mean and excludes everything I DON'T. Any help is appreciated.