I'm looking for the "most general" case in which the following statement is true:
Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be ordered sets and $f\colon \mathcal{F}_1\to\mathcal{F}_2$ an injective function, then $f$ is strictly monotonic.
I'm well aware that "the most general" isn't well defined. I'm wondering what are some (general) conditions that could be added so that the statement is true.
A couple of cases in which it's not true:
An injective real valued function of a real variable that is not monotonic on any interval.
$$\begin{align}\mathbb{R}&\to \mathbb{R}\\
x&\mapsto\begin{cases}x, &x \text{ is rational}\\
-x, & x \text{ is irrational}\end{cases}\end{align}$$
An injective continuous function that is not monotonic.
$$\begin{align}\mathbb{Q}&\to\mathbb{Q}\cup\sqrt{2}+\mathbb{Q}\\
x&\mapsto \begin{cases}x,& x< \frac{\sqrt{2}}{2},\\-x+\sqrt{2}, & x>\frac{\sqrt{2}}{2}.\end{cases}\end{align}$$
where $\mathbb{Q}$ are the rational numbers.
Edit: This question was about a function on an ordered field but, as suggested by Henry Davii on his answer, there is no need for $\mathcal{F}$ to be fields, they could just be sets. I've changed all the mentions of the word "field" on this question to "set".