Let $f$ and $g$ two analytic function. The two funtion are not equal in the whole complex plane. Due to the analytic continuation principle, any equation of the form $f(z)=g(z)$ cannot holds in any set with an accumulation point, otherwise these functions are equal for all values. Since every point in an open set is an accumulation point, then this equality must hold in a closed set. I have
$f(z)=g(z)$ in the set: $0<Re(z)<1$ and $Im(z)∈ℝ$.
Can I deduce that this hapen only on one line $Re(z)=a$ with $0<a<1$ or there are several lines in which this equality holds.