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Part CCSIR NET December 2024a-travelling-bump-of-constant-height

A travelling bump of constant height

For each positive integer n, define fn:[0,1]→Rf_{n} : [0, 1] \to \mathbb{R} by fn(x)=f_{n}(x) = nx(1−x)n(1 - x)^{n}. Which of the following statements are true?

  1. A.(fn)n≥1(f_{n})_{n}\ge_{1} does not converge pointwise on [0, 1].
  2. B.(fn)n≥1(f_{n})_{n}\ge_{1} converges pointwise to a continuous function on [0, 1].✓ Correct
  3. C.(fn)n≥1(f_{n})_{n}\ge_{1} converges pointwise to a discontinuous function on [0, 1].
  4. D.(fn)n≥1(f_{n})_{n}\ge_{1} does not converge uniformly on [0, 1].✓ 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 pointwise vs uniform.

See pricing

50 are analysed free — try those first.

The trap it tests

Pointwise vs uniform

The mode of convergence — or of continuity — was the wrong one.

Drill statements like this

Related counterexample: fₙ → f uniformly ⇒ fₙ′ → f′

More on this topic

The chapter behind this: Uniform convergence — what it buys and how to test it — free to read

From Sequences and Series of Functions › Pointwise vs uniform convergence, M-test, Dini

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

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