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Calculus Level 5

Let $f : \mathbb{C} \to \mathbb{C}$ be holomorphic on $\mathbb{C}$ .
Also, suppose the following holds: $\big|f(z)\big| \to +\infty \text{ as } |z| \to +\infty.$

Then which of the following necessarily holds?

Details and Assumptions:

$\text{Image}(f)$ denotes the range of $f.$
$A \setminus B = \{x \in A: \, x \notin B\}.$
If more than one statement is necessarily true, choose the strongest.

None of the others

f

is a polynomial

f

is bijective

\mathbb{C} \setminus \text{Image}(f)

contains at most one point

f

is surjective

f

is injective

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