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

The attained endpoint blocks continuity

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

A.f is monotone.
B.f is continuous but not strictly monotone.
C.f is not continuous.✓
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.

50 are analysed free — try those first.

Related counterexample: Continuous on a bounded interval ⇒ bounded

More on this topic

extension to closed intervalJune 2023
injective continuous implies monotoneDecember 2023
periodic uniform continuityDecember 2023
types of discontinuityDecember 2023
cauchy is preserved because the line is completeDecember 2024
a growth condition does not tame oscillationDecember 2024

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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