Part CCSIR NET June 2025a-bounded-singularity-is-removable-so-the-puncture-is-not-essential

A bounded singularity is removable so the puncture is not essential

Let 𝔻^× = { $z \in C : 0 <$ |z| < 1} be the punctured unit disk and f be a bijective holomorphic map of 𝔻^× onto itself. Which of the following statements are true?

A. $l i m_{z \to 0}$ f(z) does not exist.
B. $l i m_{z \to 0}$ f(z) exists and has absolute value ≤ 1.✓
C. $l i m_{z \to 0}$ f(z) = 0.✓
D.There exists $θ \in R$ such that f(z) = $e^{i θ} z$ for all z ∈ 𝔻^×.✓

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

A bounded singularity is removable so the puncture is not essential

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