This is a variant of question Show that if f is analytic in $|z|\leq 1$, there must be some positive integer n such that $f(\frac{1}{n})\neq \frac{1}{n+1}$.
(i). Show that if $f$ is analytic in the unit disc $D=\{z\in C : |z|<1\}$, then there exists an integer $n$ such that $f(1/n)\neq1/(n+1)$
(ii) Does there exist an analytic function $g$ in the unit disc $D=\{z\in C : |z|<1\}$ such that $g(1/n)=g(-1/n)=1/n^2$ for all positive integers $n$?
(iii) Does there exist an analytic function $g$ in the unit disc $D=\{z\in C : |z|<1\}$ such that $h(1/n)=h(-1/n)=1/n^3$ for all positive integers $n$?
Thoughts thus far: For (i) the linked problem seems to be the most concise way to show this, but we are noy concerned with the point $z = -1$ as it was in the linked problem so the.same method cannot be used as that in the linked problem For (ii) and (iii), isn't the answer no because there is a singularity at the origin. However, perhaps I am missing some lemma not shown in class that shows an analytic function may have a removable discontinuity (but I doubt this is the case).
Thank you in advance for any help that you may provide.