Thanks for this nice and explicit example, Jeremy. I wonder if there are other elementary functions besides quotients of linear functions which can be used to piece together a monotonic bijection (that there are a lot of other, but maybe not elementary, functions follows from the comments by Emil Jeřábek).

Enlightening constructions, Emil, thank you very much. So even if the rationals have not such a "smooth" structure as the reals, there is still a large degree of flexibility in reparameterizing stochastic models having a rational parameter. Have to think about the implications...

You're right, I'm not interested in these (at least not at the moment), so I should add monotonic as a requirement. Can you please provide or point to a non-monotonic continuous bijection from $\mathbb{Q}$ to itself? Thanks a lot in advance!

First of all, thanks a lot for your quick and enlightening answers! Also the discussion about intensional and extensional definitions raises an important point for further contemplation. As Francois has correctly remarked, in my context the intensional view seems to be appropriate, because I deal with programs executed on a universal reference TM. It is fascinating, that these large cardinals, which are directly concerned with unfathomable infinities, have such interesting implications for effective objects.