Putnam 2008 B5

Find all continuously differentiable functions $f: \mathbb{R} \to \mathbb{R}$ such that for every rational number $q$, the number $f(q)$ is rational and has the same denominator as $q$. (The denominator of a rational number $q$ is the unique positive integer $b$ such that $q = a/b$ for some integer $a$ with $\mathrm{gcd}(a,b) = 1$.) (Note: gcd means greatest common divisor.)