Well we couldn't get the same nonnegative rational number twice. There would have to be a first such number (call it \\(x\\)), and it would have to be reachable from 2 different values \\(f(y) = x, f(z) = x, y \neq z\\) which is impossible because \\(f\\) is injective, with inverse \\( f^{-1}(x) = 2\lceil 1/x-1 \rceil - (1/x-1) \\), and \\( f^{-1}(0) \\) is undefined.