In Lie algebras, and representations an elementary introduction by Hall B. it is mentioned that: "$\exp(\log(z))=z$ for $z\in (0,2)$ ($z$ real) and since both sides of this identity are holomorphic in $z$, the identity continues to hold on the whole set $\left\{|z-1|<1\right\}$."
From what I see, the argument is as follows
If $f(x)=0$ for $x\in (-R,R)$ with $f(x)$ holomorphic then $f(z)=0$ for $z\in \left\{z\in\mathbb{C}:|z|<R\right\}$?
Why can this function be extended to a complex variable? is it part of some theorem?