I got the following question from my complex analysis class ($\hat{\Bbb{C}}$ denotes the Riemann-sphere):
Does there exist a holomorphic function $ f:\Bbb{C} \to \hat{\Bbb{C}}$ such that $f(n)=\infty$ for all $n \in \Bbb{N}$?
I would say that, since it's not specified that the function must not be constant, we can take $f(z) = \infty$ for all $z\in\Bbb{C}$, though I'm a little rusty in checking holomorphicity so I'm not sure whether it actually is holomorphic.
My other question is: is there any other holomorphic function satisfying this property? Perhaps $f(z) = \prod_{n \in \Bbb{N}} \frac{1}{z-n}$ is holomorphic on $\hat{\Bbb{C}}$?
My worry for this function is that for the Identity Theorem on $\hat{\Bbb{C}}$, since $\{f=\infty\}$ has an accumulation point in $\hat{\Bbb{C}}$, namely $\infty$, would imply that $f=\infty$ everywhere. But I'm not even sure whether $\infty$ is indeed an accumulation point in $\hat{\Bbb{C}}$.
Thanks in advance