I'm curious how this applies to another sort of edit, the "rebasing" operations, for instance G(X) := replace 2^n in the binary expansion of X with 3^n. G maps rationals into rationals, the unit interval into itself, preserves ordering and so on.
(5) Is F(X) continuous? (I think yes. Certainly if X and Y are both rational and |X-Y|-->0, then |F(X)-F(Y)|-->0.)
Also true for G, I guess? However there are an awful lot of "holes" in the range -- every number that contains a 2 anywhere in its ternary expansion.
There also are interesting variants of this construction
I suppose we could hack up algebraic variants by editing representations of algebraic numbers in analogous ways, for instance twiddling the coefficients of the polynomial that X is the root of.
I'm not sure whether these are continuous functions of X
Heck, I'm not even sure I should continue to believe in the reals.