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Part BCSIR NET June 2024vacuous-hypothesis

Vacuous hypothesis

Let f be an entire function. Which of the following statements is FALSE?

  1. A.If Re(f), Im(f) are bounded then f is constant
  2. B.If e^(|Re(f)| + |Im(f)|) is bounded, then f is constant
  3. C.If the sum Re(f) + Im(f) and the product Re(f)Im(f) are bounded, then f is constant
  4. D.If sin(Re(f) + Im(f)) is bounded, then f is constant✓

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.

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50 are analysed free — try those first.

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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