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Calculus Level 5

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

Then which of the following necessarily holds?


Details and Assumptions:

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