Is the mapping $$ f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n} $$ surjective?
If not, what is its image?
If yes, what can be said about images of intervals, besides the obvious $f([-1,1]) = \{0,\frac{2}{3},1\}$?