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Part BCSIR NET December 2025composition-of-two-uniformly-continuous-functions-is-uniformly-continuous-in-both-orders

Composition of two uniformly continuous functions is uniformly continuous in both orders

Let f:(0,∞)→(0,∞)f : (0,\infty) \to (0,\infty) be a uniformly continuous function. Define g:(0,∞)→(0,∞)g : (0,\infty) \to (0,\infty) by g(x)=x2g(x) = x^{2} if 0 < x ≤ 1, and g(x) = x if x > 1. Which of the following statements is true?

  1. A.f∘g is uniformly continuous, but g∘f is not uniformly continuous.
  2. B.g∘f is uniformly continuous, but f∘g is not uniformly continuous.
  3. C.Neither f∘g nor g∘f is uniformly continuous.
  4. D.Both f∘g and g∘f are 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.

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50 are analysed free — try those first.

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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