Suppose I have $x\in \mathbb{Q}\cap [0,1]$, where $\mathbb{Q}$ is the set of rational numbers, and let $|x|$ be the length of $x$ when expressed in binary. I wish to find an analytic function of $x$ which interpolates the points $(x,|x|)$.
Is this possible, and if so is this function computable? Since $\bar{\mathbb{Q}}\cap [0,1]$ is a countable number of points it seems to satisfy some of the necessary properties for this scheme.