Is $f(D)$ is open /closed? Consider  the  polynomial 
$$
f(z)  = \sum_{n=1}^{N} a_n z^n\quad 1 \le N < \infty,\: a_n  \in  \mathbb{R} \setminus\{0\}
$$
and consider  $D = \{ w  \in \mathbb{C} : |w| < 1\}$
Choose the correct option


*

*$f(D)$ is closed.

*$f(D)$ is open.
My try
By  Liouville's theorem  $f(D)$  will be constant so option 1)  will true that f(D)  will be closed 
Am I right??

Louiville's Theorem is for analytic functions  bounded on the entire complex plane . The present function is not a constant. The correct argument is to use open mapping theorem which says that a non-constant analytic function maps open sets to open sets.  So the correct option is 2).
