How can one characterize monotonic bijections from $\mathbb{Q}$ to
$\mathbb{Q}$? It is easy to see that piecewise linear functions which are
strictly monotonic and surjective will do the trick, but are these functions
already all monotonic bijections of the rationals, or are there also "curved"
ones? Classical curved bijections of the reals like $x^3$ are of no help,
because, for example, 2 has no preimage in $\mathbb{Q}$.


The context of this question is the transformation of random variables
and the degrees of freedom associated with these transformations, or more specific,
what kind of results valid for continuous random variables can be transferred
to rational random variables, and which results are sensitive to the special
structure of the reals.



Monotonic bijections of rational numbers
A simple example of a monotonic homeomorphism from $\mathbb{Q}$ to itself that is not piecewise linear is given by $f(x)=x$ if $x<0$, $f(x)=\frac{x}{1-x}$ if $x\in[0,\frac{1}{2}]$ and $f(x)=x+\frac{1}{2}$ if $x>\frac{1}{2}$.


