Before I start a little disclaimer:
I am a little bit new to the concept of o-minimal structures, so my apologies in advance if this question is a bit on the trivial side of things. If so, could you perhaps point me towards some resources that might help answer these kind of questions? Until now, I have noticed that resources like "Tame Topology and O-Minimal Structures" by L. P. D. van den Dries seem to do the following:
- Give some definition of definability (or actually some definition of o-minimal structures)
- List some properties that these definable functions (and sets) enjoy. One of which being for example "Monotonicity Theorem" that roughly says that every definable is piecewise constant/strictly monotonic.
For the research I am currently doing, I tend to want to know the opposite. I'd like to know given some function $\mathbb{R} \to \mathbb{R}$ that is strictly monotonic, whether that is definable. Might it be true for all strictly monotonic functions?
Thank you in advance for your help!
That is not true in general. You can take a zigzag function $f:\mathbb{R} \to \mathbb{R}$ that is strictly monotonic with a line that intersects the graph of $f$ at an infinite number of points. Then, clearly, $f$ is not definable in any o-minimal structure on $(\mathbb{R},+,\cdot)$. Indeed, fix an o-minimal structure on $(\mathbb{R},+,\cdot)$, if $f$ is definable (meaning the graph of $f$ is a definable subset of $\mathbb{R}^2$), then also the intersection of the graph of $f$ with the line is definable, but this is not possible (because any definable set has only a finite number of connected components). I hope this answers your question.
For references, I would recommend "Tame Topology and O-Minimal Structures" by L. P. D. van den Dries, and "Introduction to o-minimal geometry" by M. Coste (https://perso.univ-rennes1.fr/michel.coste/polyens/OMIN.pdf).