This question already has an answer here:
Let $f$ be a function which is continuous on the closed unit disc and analytic on the open disc. Assume that $|f(z)|=1$ whenever $|z|=1$. Show that the function $f$ can be extended meromorphically to the whole complex plane with at most a finite number of poles.
I think it feasible to construct such a meromorphic function but how, and is it possible to avoid such constructions?