I am looking for as general a class as possible of real functions defined on $\mathbb{R}^+$ that are guaranteed to have a finite number of zeroes - no, polynomials are not enough :).


Specifically, consider a class of functions defined like [elementary functions](http://en.wikipedia.org/wiki/Elementary_function), but without allowing for complex constants. It seems to me that such functions must have a finite number of zeros. Any ideas on how to prove this (or counterexmamples)?




---


*Background*: What I'm actually looking to do is prove that a real postive function $f:\mathbb{R}^+\to\mathbb{R}^+$ is eventually concave, i.e. there exist $x\_0\geq 0$ such that $f(x)$ is concave for every $x\geq x\_0$. I know that $f$ is real analytic, increasing and upper bounded, but its exact formula is intractable. It thus suffices to show that $f''$ has a finite number of zeroes and hence my question. The structure of $f''$ is more complicated than a "limited elementary function" described above, but a result on such functions will definitely be a step in the right direction.



Real functions with finitely many zeroes
Let $\mathcal R$ be any [o-minimal](http://en.wikipedia.org/wiki/O-minimal_theory) expansion of the real ordered field $(\mathbb R,<,0,1,+,-,\cdot)$, and $\mathcal F$ be the class of functions (first-order) definable (with real parameters) in $\mathcal R$. On the one hand, $\mathcal F$ has various nice closure properties (in particular, it is closed under composition, taking inverse functions, and derivatives). On the other hand, o-minimality guarantees that for any $f\in\mathcal F$, $f\colon\mathbb R\to\mathbb R$, its positive set $\{x\in\mathbb R:f(x)>0\}$ is a finite union of points and intervals; in particular, $f$ is eventually positive, eventually negative, or eventually constant $0$.


Note that in practice, theories of structures known to be o-minimal are often also [model complete](http://en.wikipedia.org/wiki/Model_complete), hence a function is definable iff its graph is a projection of a Boolean combination of positive sets of the basic functions included in its signature.


Wilkie proved that the exponential field $\mathbb R\_{\exp}=(\mathbb R,\exp)$ is o-minimal. The class of functions definable in $\mathbb R\_{\exp}$ includes the functions mentioned in your question, so the answer to your specific question is positive.


Even larger expansions of $\mathbb R$ are known to be o-minimal. First, by a result of van den Dries, $\mathbb R\_\mathrm{an}$ is o-minimal, which is the expansion of $\mathbb R$ by all real-analytic functions $f\colon[0,1]^n\to\mathbb R$ (extended by the constant $0$ function outside $[0,1]^n$ to be defined on the whole of $\mathbb R^n$). Second, the pfaffian closure $\mathcal R\_\mathrm{pfaff}$ of any o-minimal expansion $\mathcal R$ of $\mathbb R$ is again o-minimal, due to Speisseger. In particular, $\mathbb R\_\mathrm{an,pfaff}$ is o-minimal. (The full definition of the pfaffian closure can be found e.g. in [1]. In particular, it includes all [pfaffian functions](http://en.wikipedia.org/wiki/Pfaffian_function) such as $\exp$.)


[1] Patrick Speissegger, *Pfaffian Sets and O-minimality*, in: Lecture Notes on O-Minimal Structures and Real Analytic Geometry (C. Miller, J.-P. Rolin and P. Speissegger, eds.), Fields Institute Communications vol. 62, 2012, pp. 179–218, <http://dx.doi.org/10.1007/978-1-4614-4042-0_5>


