= Identity theorem
{wiki}
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.