# Meromorphic continuation of function analytic on the open unit disc [duplicate]

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?

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• We can show that $f$ is a finite Blaschke product. It can be extended meromorphically with at most finite number of poles. – Song Jan 2 at 11:40