Part CCSIR NET December 2023identity-theorem-pigeonhole

Identity theorem pigeonhole

Let X be an uncountable subset of $\mathbb{C}$ and let $f : \mathbb{C} \to \mathbb{C}$ be an entire function. Assume that for every z ∈ X there exists an integer n ≥ 1 such that $f^{(n)}(z)$ = 0. Which of the following statements are necessarily true?

A.f = 0.
B.f is a constant function.
C.There exists a compact subset K of $\mathbb{C}$ such that $f^{-1}(K)$ is not compact.
D.f is a polynomial.✓ 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.

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

Identity theorem pigeonhole

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