the identity theorem
How much of an analytic function do you need to know to know all of it? Astonishingly little. The identity theorem says that an analytic function on a connected region is pinned down completely by its values on any set that has a limit point inside the region — even just a single convergent sequence of points. If two analytic functions agree on such a tiny set, they agree everywhere. There is no freedom to differ later: the local data already determines the global function.
Precisely: let f and g be analytic on a connected open set D. If the set of points where f = g has a limit point (an accumulation point) inside D, then f = g throughout D. The proof is the isolation of zeros in disguise. Consider h = f - g, which is analytic and vanishes on a set with a limit point. If h were not identically zero, its zeros would be isolated — but a limit point of zeros contradicts isolation. So h is identically zero on a neighbourhood, and a connectedness argument spreads this 'identically zero' over all of D. Connectedness is essential: on two separate pieces, f could equal g on one and differ on the other.
This is the source of complex analysis's uniqueness and rigidity. It means a holomorphic function has no local wiggle room: you cannot perturb it on a tiny patch and keep it analytic. It guarantees that a real function (like sin x or e^x) has at most one analytic extension to the complex plane — so the complex sine and exponential are forced, not chosen. And it is the logical backbone of analytic continuation: if you can extend a function past its original domain at all, the extension is unique, because any two extensions agree on the overlap and hence everywhere.
Two entire functions agree at z = 1, 1/2, 1/3, ... which accumulate at 0; therefore they agree everywhere. This is how we know there is only ONE entire function equal to sin x on the real axis: the reals have limit points, so any analytic extension is forced to be the usual complex sin z. Identities like sin^2 z + cos^2 z = 1 transfer from the real line to the whole plane for the same reason.
Agreement on a sequence with a limit point inside the region forces agreement everywhere — local data fixes the whole function.
The limit point must lie INSIDE the connected region, and the region must be connected. Agreement on a set whose only limit point is on the boundary (like the integers, which only accumulate at infinity) does not force the functions to coincide.