Given 2 analytic functions $ f(x) $ and $ g(x) $

Question

let be 2 analytic function so in the real line for every real number 'x' then

$$ f(x)=g(x) $$

then can we assure that for every 'z' complex number can we also have

$$ f(z)=g(z) $$

so if 2 functions are equal for are the real numbers then they are also equal for every complex number, if they ara analytic whitout poles

Answer 1

True because of the identity theorem (obviously, the set $\Bbb R$ has a cluster point in $\Bbb C$).