$f$ entire and non-constant function, then for fix $x \in \mathbb{C}$ the set of $(z \in \mathbb{C} : f(z)=x)$ is finite or countable.
14117
Holomorphic function has at most countably zeros
32774
Finite Cartesian Product of Countable sets is countable?
94190
Disproving the statement "If $f(z)$ is not an entire function, then $g(z) = f^2(z)$ is not entire.”
148992
A finite set is closed
33157
Proof that an infinite subset of $\mathbb{N}$ is countable
68071
If $f$ is a nonconstant entire function such that $|f(z)|\geq M|z|^n$ for $|z|\geq R$, then $f$ is a polynomial of degree atleast $n$.
123467
Prove that $f(z) = e^z$ maps $\mathbb{C} \setminus B $ onto $\mathbb{C} \setminus \{0\}$.
98151
Suppose that A is a countable set. Show that the set B is also countable if there is an onto function f from A to B.
135577
Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
45219
Proving that $C$ is a subset of $f^{-1}[f(C)]$
8179
If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
9021
An infinite subset of a countable set is countable
36379
Suppose that $ f $ is entire and that for each $ z $, either $ |f(z)| \leq 1 $ or $ |f^\prime (z) |\leq 1 $. Prove that $ f $ is a linear polynomial.
28011
Does there exists an entire function such that for each $|z|>C$ for some constant $C$, $|f(z)|=|z|+1$?
63678
Let $A$ and $B$ be countable sets. Is there any function $f$ such that a certain condition holds for an uncountable number of functions $g$?
98399
Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.
105700
Proving $\,f$ is constant.