I'm having trouble with the following question(s) related to analytic functions. We haven't learned about Picard's Theorems in class (which seems to be related to this question), and I'm not sure how to figure it out without using it. The question is as follows:
Why is it true that an entire function $f(z)$ which does not send any $z \in \mathbb{C}$ to half of the plane must be constant?
Would it also be true that an entire function which does not send any $z \in \mathbb{C}$ to an arc must be constant as well?