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Part CCSIR NET December 2023injective-continuous-implies-monotone

Injective continuous implies monotone

Let f:R→Rf : \mathbb{R} \to \mathbb{R} be a continuous function such that |f(x) − f(y)| ≥ log(1 + |x − y|) for all x,y∈Rx, y \in \mathbb{R}. Which of the following statements are true?

  1. A.f is necessarily one-one.✓ Correct
  2. B.f need not be one-one.
  3. C.f is necessarily onto.✓ Correct
  4. D.f need not be 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, and why this one tests converse assumed.

See pricing

50 are analysed free — try those first.

The trap it tests

Converse assumed

The theorem runs one way. You used it in the other.

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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