Let $f: A \to \mathbb{R}$ a non-constant (and not necessarily analytic) continuous function, where $A \subset \mathbb{R}$ is an open set. $f$, as well as its derivative, $f'$, blow up at the boundary of $A$. However, in any compact subset of $A$, let B, $f'$ is bounded. Is there anything I can say about the distance of consecutive zero crossings of $f$ in B ?
Optimally, I want to say that the difference $b-a$ of any two points $a$, $b$ such that $f(a)=f(b)=0$, and $f(x) \neq 0, \forall x\in(a,b)$ (i.e., two consecutive zero-crossings), is greater than a positive constant. - Which, for instance, doesn't happen with functions like $\sin(\frac{1}{x})$ or $\sin(e^x)$. If not, is there a counterexample ?
No, you can't. Take any sequence $a_n$ with a limit $a$. You can use a series of smooth "bump functions" to construct smooth functions that have zero crossings at all the $a_n$. You can even get zero crossings on an arbitrary closed nowhere-dense set, e.g. the Cantor set (for an appropriate definition of "zero crossings").