[Math] Let $S$ be the open unit disk and $f: S\to \Bbb C$ be a real-valued analytic function with $f(0)=1$

Question

I have this problem:

Let $S$ be the open unit disk in $\mathbb C$ and $f:S\to \Bbb C$ be a real-valued analytic function with $f(0)=1$.Then which of the following option is correct?

The set $\{z \in S:f(z) \neq 1\}$ is:

(a) empty,

(b) non-empty finite,

(c) countably infinite,

(d) uncountable.

Please help.

Answer 1

Hint: Since $f$ is analytic it's also holomorphic. Using the Riemann-Cauchy equations one can show that a real-valued holomorphic function (on a domain) has to be constant.