Part CCSIR NET June 2024monotone-does-not-mean-onto

Monotone does not mean onto

Let $f : R \to R$ be a continuous and one-to-one function. Which of the following statements are necessarily true?

A.f is strictly increasing
B.f is strictly decreasing
C.f is either strictly increasing or strictly decreasing✓
D.f is onto

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
the attained endpoint blocks continuityDecember 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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