This is exercise 5, page 236 from Remmert, Theory of complex functions
For each of the following properties produce a function which is holomorphic in a neighborhood of $ 0 $ or prove that no such function exists:
i) $ f (\frac{1}{n}) = (-1)^{n}\frac{1}{n} \ $ for almost all $ n \in \mathbb{N}\ , n \neq 0 $
ii) $ f (\frac{1}{n}) = \frac{1}{n^{2} - 1 } $ for almost all $ n \in \mathbb{N}\ , n \neq 0, n \neq 1 $
iii) $|f^{(n)}(0)|\geq (n!)^{2} $ for almost all $ n \in \mathbb{N} $