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Part CCSIR NET December 2024omitting-a-disc-invites-liouville-on-the-reciprocal

Omitting a disc invites liouville on the reciprocal

Let f:Cf : \mathbb{C} \ {−1, 1} →C\to \mathbb{C} be a holomorphic function that does not take any value in the set {z∈Cz \in \mathbb{C} : |z − 1| < 1}. Which of the following statements are true?

  1. A.f is constant.✓ Correct
  2. B.f has removable singularities at −1 and 1.✓ Correct
  3. C.f is bounded.✓ Correct
  4. D.f has either poles or essential singularities at −1 and 1.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests hypothesis dropped.

See pricing

50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Related counterexample: A bounded holomorphic function on an unbounded domain is constant

More on this topic

The chapter behind this: Liouville, Morera and the maximum principle — free to read

From Cauchy Theory › Liouville, Morera, maximum modulus principle

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