 Analytic continuation

An analytic continuation extends a holomorphic function through overlapping connected open sets while preserving its values on their overlap. Continuation along different paths can produce different germs when the domain contains branch points.

visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) is a good quick visual non-mathematical introduction is to it.

The key question is: how can this continuation be unique since we are defining the function outside of its original domain?

The answer is: due to the identity theorem.