TITLE: Singularity at infinity of a function entire
QUESTION [14 upvotes]: How to prove that every non-constant entire function $\,\,f:\mathbb{C}\rightarrow\mathbb{C}\,\,$ has a singularity at infinity?
What type of singularity must this be?

REPLY [5 votes]: Just to give an example.
Clearly $f$ is unbounded near $\infty$ so the singularity cannot be removable. 
If $f(z)=az+b$ where $a\neq 0$ , then $f$ has a pole at $\infty$.
If $f(z)=e^z$, then $f$ has an essential singularity at $\infty$.