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Part CCSIR NET June 2024expansive-map-inverts-to-lipschitz

Expansive map inverts to lipschitz

Let K⊆RK \subseteq \mathbb{R} be non-empty and f : K → K be continuous such that |x − y| ≤ |f(x) − f(y)| for all x, y ∈ K. Which of the following statements are true?

  1. A.f need not be surjective✓ Correct
  2. B.f must be surjective if K = [0, 1]✓ Correct
  3. C.f is injective and f⁻1:f(K)→K^{1} : f(K) \to K is continuous✓ Correct
  4. D.f is injective, but f⁻1:f(K)→K^{1} : f(K) \to K need not be 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 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

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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