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Part CCSIR NET June 2024uniform-on-every-compact-piece-is-not-uniform-overall

Uniform on every compact piece is not uniform overall

Let f:[0,1)→[1,∞)f : [0, 1) \to [1, \infty) be defined by f(x) = 1/(1 − x). For n ≥ 1, let pn(x)=1+x+⋯+xnp_{n}(x) = 1 + x + \cdots + x^{n}. Then which of the following statements are true?

  1. A.f(x) is not uniformly continuous on [0, 1)✓ Correct
  2. B.The sequence (pn(x))(p_{n}(x)) converges to f(x) pointwise on [0, 1)✓ Correct
  3. C.The sequence (pn(x))(p_{n}(x)) converges to f(x) uniformly on [0, 1)
  4. D.The sequence (pn(x))(p_{n}(x)) converges to f(x) uniformly on [0, c] for every 0 < c < 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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