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

A growth condition does not tame oscillation

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

  1. A.S is empty.
  2. B.S is non-empty.✓ Correct
  3. C.f must be uniformly continuous.
  4. D.f need not be uniformly continuous.✓ Correct

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 hypothesis dropped.

See pricing

50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

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.

ShareWhatsAppTelegram