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Part BCSIR NET December 2024the-attained-endpoint-blocks-continuity

The attained endpoint blocks continuity

Let f:R→[0,∞)f : \mathbb{R} \to [0, \infty) be a bijective function. Which of the following statements is true?

  1. A.f is monotone.
  2. B.f is continuous but not strictly monotone.
  3. C.f is not continuous.✓ Correct
  4. D.f is continuous but not uniformly continuous.

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 boundary and endpoint.

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50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Related counterexample: Continuous on a bounded interval ⇒ bounded

More on this topic

The chapter behind this: Continuity vs uniform continuity vs Lipschitz — free to read

From Continuity and Differentiation › Continuity, uniform continuity, Lipschitz

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