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Part BCSIR NET June 2025continuous-limit-does-not-make-the-convergence-uniform

Continuous limit does not make the convergence uniform

For each n ≥ 1, let fn:[0,1]→Rf_{n} : [0, 1] \to \mathbb{R} be defined as fn(x)=f_{n}(x) = nx if x∈[0,1/n],fn(x)=2−x \in [0, 1/n], f_{n}(x) = 2 - nx if x ∈ (1/n, 2/n), and fn(x)=0f_{n}(x) = 0 if x ∈ [2/n, 1]. Which of the following statements is true?

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

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

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