Which of the followings are correct:
My thought: $g(z)=\dfrac{2}{3+z}$ is analytic on $\mathbb C-\{3\}.$ Also $f=g$ on $\{\dfrac{1}{n}\}$ which has a limit point $0$ on $\mathbb D.$ So by identity theorem $f=g$ on any domain contained in the domains of the definitions of both $f$ and $g.$
1 is true: $f(0)=\dfrac{2}{3+0}=\dfrac{2}{3};$ 4 is false: $f(z)=\dfrac{2}{3+z};$ 3 & 2 are false: $3,-3$ is outside the domain of definition of $f;$ e.g. let $f(z)=\dfrac{2}{3+z}~\forall~|z|\leq 1$ and $f(z)=1~\forall~|z|>1.$
Please tell me whether I'm right or wrong!