Denote by $\Sigma$ the collection of all $(S, \succeq)$ wher $S \subset \mathbb{R}$ is compact and $\succeq$ is an arbitrary total order on $S$.
Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ such that for all $(S, \succeq) \in \Sigma$ there exists a compact interval $I$ with the properties that
- $f(I) = S$
- $x \geq y$ implies $f(x) \succeq f(y)$ for all $x,y \in I$?
If so, how regular can we take $f$ to be? The motivation is that basically, I am trying to construct the analogue of a normal sequence but on $\mathbb{R}$ instead of $\mathbb{N}$.
EDIT: As Brian M. Scott points out, this is not possible if the orderings have no greatest and least elements. However, since adding this assumption doesn't go against the intuition of generalizing normal sequences, I am still interested in the answer if we restrict the various total orders to have minimal and maximal elements.
Thanks in advance.