Allan - I doubt \\( f(x) = x\|I_Q(x) - 1\|\\) is an addition-preserving function from the reals to the reals, because the only such functions for which an explicit formula is possible are the functions \\(f(x) = c x \\). The rest require the Axiom of Choice, as mentioned in an earlier comment.

What happens with \\(f(x+y) = f(x) + f(y)\\) when \\(x,y\\) are irrationals that sum to a rational?

What happens with \\(f(x+y) = f(x) + f(y)\\) when \\(x,y\\) are irrationals that sum to a rational?