How can one characterize continuous bijections from $\mathbb{Q}$ to $\mathbb{Q}$? It is easy to see that piecewise linear functions which are monotone, continuous and surjective will do the trick, but are these functions already all continuous 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.

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.