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

Expansive map inverts to lipschitz

Let $K \subseteq R$ 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?

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

See pricing

50 are analysed free — try those first.

Related counterexample: Continuous on a bounded interval ⇒ bounded

Expansive map inverts to lipschitz

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