  We carry out an arithmetical study of analytic functions $f: [0,1] \to [0,1]$
that by restriction induce a bijection $\mathbb{Q} \cap [0,1] \to \mathbb{Q}
\cap [0,1]$. The existence of such functions shows that, unless $f(x)$ has some
additional property of an algebraic nature, very little can be said about the
distribution of rational points on its graph. Some more refined questions
involving heights are also explored.
