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Part CCSIR NET December 2024integration-smooths-but-does-not-invert

Integration smooths but does not invert

Let f:[0,1]→Rf : [0, 1] \to \mathbb{R} be a Riemann integrable function. Define F(x)=∫0F(x) = \int_{0}ˣ f(t)dt, ∀x ∈ [0, 1]. Which of the following statements are necessarily true?

  1. A.F is Riemann integrable.✓ Correct
  2. B.If F(x) = 0 for all x ∈ [0, 1], then f(x) = 0 for all x ∈ [0, 1].
  3. C.F is uniformly continuous on [0, 1].✓ Correct
  4. D.F is differentiable 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 converse assumed.

See pricing

50 are analysed free — try those first.

The trap it tests

Converse assumed

The theorem runs one way. You used it in the other.

Drill statements like this

Related counterexample: |f| Riemann integrable ⇒ f Riemann integrable

More on this topic

The chapter behind this: Riemann integrability — the criterion and the zoo — free to read

From Integration › Riemann integration and criteria

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