Identity Theorem: Extending from $\mathbb{R}$ to $\mathbb{C}$

Question

Suppose we have $f_1, f_2: \mathbb{C} \rightarrow \mathbb{C}$ holomorphic, and $f_1 = f_2$ on $\mathbb{R}$. Can we then say $f_1 = f_2$ identically on $\mathbb{C}$?

This appears to be true by the uniqueness of analytic continuations, but the identity theorem requires that the functions must agree on an open set, and clearly $\mathbb{R}$ is not open with respect to $\mathbb{C}$.

Assuming the above is true, why can this not be shown using the identity theorem?

Answer 1

The identity theorem does not require that the functions must agree on an open set !

Let $A:=\{z \in \mathbb C:f_1(z)=f_2(z)\}$ .

If $A$ has an accumulation point in $ \mathbb C$, then , by the identitiy theorem, $f_1=f_2$ on $ \mathbb C$.

In your case we have $A = \mathbb R$.

Answer 2

Yes, it is true. For the identity theorem, it is enough to assume that $f_1 = f_2$ on a set with at least one accumulation point.