TITLE: Let $f$ be an entire function and $L$ a line in $\mathbb{C}$ such that $f(\mathbb{C})\cap L=\emptyset$. Show that $f$ is constant function.
QUESTION [6 upvotes]: Let $f$ be an entire function and $L$ a line in $\mathbb{C}$ such that $f(\mathbb{C})\cap L=\emptyset$. Show that $f$ is constant function.
If $f$ is not constant  then $f(\mathbb{C})$ is dense set in $\mathbb{C}$, but how can I use that line does not intersect the image set?

REPLY [2 votes]: As $f$ is entire and non-constant, $f$ takes all values with atmost one exception possibly (Little Picard's theorem). But $f(\mathbb C)\cap L=\emptyset$ implies $f$ skip infinite points on $\mathbb C$, so $f$ must be constant.