Uniqueness theorem

Will anybody please tell me what is the statement of the "Uniqueness theorem" in Complex analysis??

 

I have the sum:
Let f:C-->C be analytic s.t f(1/n)=1/n^2,n=1,2,3........
Then(out of 4 ans i am giving the correct one)f(z)=z^2 for all z in C
The solution is done as follows:
Let g(z)=z^2 for all z in C.Then f(z)=g(z) for all z in {1/n :n=1,2,3.}.since {1/n :n=1,2,3.} is an infinite set having a limit point "0",so by "uniqueness theorem" f(z)=g(z) for all z in C.

Here i want to have the statement of this Uniqueness theorem...