Well, it can be shown by induction of each of the parts that any number that can be formed by a finite number of additions, subtractions, multiplications, divisions, and square roots, applying the said operations in any order, then f fixes this number. So that covers quite a few numbers, but I really doubt that every real number can be expressed in this way. I could prove that by showing the set of all finite length algebraic expressions using sums, differences, multiplications, quotients, and roots, to be countable. I guess that is obvious, since the set of such algebraic expressions would correspond to a subset of the language of all strings containing '( ) + * - / ^ 1' which is a countable set.
So no, I don't have the reals yet. I think continuity is needed. I'll try to produce a counter example.