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Part CCSIR NET June 2025countably-many-isolated-zero-sets-cannot-cover-an-uncountable-plane

Countably many isolated zero sets cannot cover an uncountable plane

Let f be an entire function which is not a polynomial. Let A = {α∈C\alpha \in \mathbb{C} | f⁽n^{n}⁾(α)≠0(\alpha) \ne 0 for all n ≥ 0}. Which of the following statements are true?

  1. A.A is nonempty.✓ Correct
  2. B.A is finite.
  3. C.A is infinite.✓ Correct
  4. D.A is uncountable.✓ Correct

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 standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: Zeros of a non-constant holomorphic function cannot accumulate

More on this topic

The chapter behind this: Analyticity, the identity theorem and zeros — free to read

From Analytic Functions › Power series and analyticity

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