Let $f(z)$ be a power-series (with complex coefficients) centered at $0 \in \mathbb{C}$ and with a radius of convergence 2 . Suppose that $f(0)=0$. Choose the correct statement(s) from below:\
- $f^{-1}(0)=\{0\}$;
- If $f$ is a non-constant function on $\{|z|<2\}$, then $f^{-1}(0)=\{0\}$;
- If $f$ is a non-constant function, then for all $\zeta \in \mathbb{C}$ with sufficiently small $|\zeta|$, the equation $f(z)=\zeta$ has a solution;
- $$ \int_{\gamma} f^{(n)}(z) \mathrm{d} z=0 $$ for every $n \geq 1$, where $\gamma$ is a unit circle centred at 0 , oriented clockwise, and $f^{(n)}$ is the $n$th derivative of $f(z)$.
In this how to prove inverse image of a singleton set zero is not a singleton set zero? Thanks in advance.