TITLE: If $f,g$ are both analytic and $f(z) = g(z)$ for uncountably many $z$, is it true that $f = g$? 
QUESTION [21 upvotes]: If two analytical functions of $\mathbb{C}$  f and g are equal on an infinite number of input values, than they are equal. I can't seem to find a counterexample, but I haven't seen this anywhere except on the particular practice exam question I'm trying to solve. 
Edit:Uncountably infinite. Sorry about that.

REPLY [4 votes]: Any uncountable subset of Euclidean space has a limit point (in our case, we have that the set of points where f and g agree has a limit point). This is because $E^n$ is Lindelöf, i.e.,every cover of $E^n$ by open sets has a countable subcover, and by Weierstrass Theorem on Infinite Bounded Subsets of $ E^n$.
We  take a cover C by balls $B(0,n)$, for $n=1,2, \dots$, and, using Lindelöf, we take a finite subcover of C. Notice that each of the elements of this cover is bounded. 
Now, using basic set theory ( I'm no expert myself), one of these bounded balls $B:=B(0,n_0)$ must contain infinitely-many points (i.e., countable union of countable is countable). Then we can use Weierstrass' theorem that every bounded, infinite subset of $E^n$ must contain a limit point, applying it to our ball B. Then we can apply the identity theorem to the set of points where f and g agree.
{"set_name": "stack_exchange", "score": 21, "question_id": 35767}