Part CCSIR NET December 2024a-growth-condition-does-not-tame-oscillation

A growth condition does not tame oscillation

Let $f : [0, \infty) \to [0, \infty)$ be a continuous function such that $lim (x \to \infty) f (x) / x = α < 1$ . Let S = { $c \in [0, \infty)$ | f(c) = c}. Which of the following statements are true?

A.S is empty.
B.S is non-empty.✓
C.f must be uniformly continuous.
D.f need not be 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.

See pricing

50 are analysed free — try those first.

Related counterexample: Continuous on a bounded interval ⇒ bounded

A growth condition does not tame oscillation

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