the uniqueness of analytic continuation
When you continue a holomorphic function from a small patch to a larger region, you might worry there are many ways to do it, each giving a different answer in the new territory. The reassuring news is that there is essentially only one. If two holomorphic functions agree on the original patch — indeed if they agree on any little disk, or even on a curve, or on a sequence of points piling up at an interior point — then they agree everywhere on any connected region containing both. So once you fix the function on a tiny seed, the rest of the holomorphic extension is completely determined.
This is not magic; it is the identity theorem in action. The identity theorem says that if two holomorphic functions f and g on a connected domain coincide on a set with a limit point inside the domain, then f = g throughout. The reason behind it is the rigidity of holomorphic functions: their zeros are isolated, so the difference f - g, being holomorphic and vanishing on a non-isolated set, must be identically zero. To use it for continuation: suppose two continuations both agree with the original series inside the first disk. Their difference is holomorphic on the combined region and zero on a whole disk — a set jammed with limit points — so by the identity theorem the difference is zero everywhere the two are both defined. The continuations coincide.
There is one crucial caveat hiding in the word 'connected', and more precisely 'simply connected'. Uniqueness holds along any single chain of overlapping disks, but if the larger region has a hole, you can continue the SAME starting function around the hole by two different routes and arrive at two genuinely different values at the destination — the logarithm does exactly this around 0. So uniqueness is local-and-along-a-path; the global single-valuedness is a separate, more delicate question that the monodromy theorem and Riemann surfaces will settle.
Suppose a holomorphic function on the unit disk equals sin z for every real x in (-1, 1). Those points pile up (a limit point in the disk), so by the identity theorem the function must BE sin z on the whole disk, and then sin z is its unique continuation to the entire plane. There is no holomorphic function that matches sin on a real interval but disagrees with it elsewhere.
Agreeing on a set with an interior limit point forces agreement everywhere connected — the continuation is pinned down.
Uniqueness is along a path inside a connected domain; it does NOT promise a single-valued global function. Continuing around a hole by different routes can land on different branches — that is multivaluedness, not a contradiction.