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 = { $α \in C$ | f⁽ $^{n}$ ⁾ $(α) \neq = 0$ for all n ≥ 0}. Which of the following statements are true?

A.A is nonempty.✓
B.A is finite.
C.A is infinite.✓
D.A is uncountable.✓

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.

See pricing

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

Countably many isolated zero sets cannot cover an uncountable plane

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