Incoming links: Analytic continuation
Essentially, defining an holomorphic function on any open subset, no matter how small, also uniquely defines it everywhere.
This is basically why it makes sense to talk about analytic continuation at all.
One way to think about this is because the Taylor series matches the exact value of an holomorphic function no matter how large the difference from the starting point.
Therefore a holomorphic function basically only contains as much information as a countable sequence of numbers.