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

Integration smooths but does not invert

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

A.F is Riemann integrable.✓
B.If F(x) = 0 for all x ∈ [0, 1], then f(x) = 0 for all x ∈ [0, 1].
C.F is uniformly continuous on [0, 1].✓
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.

See pricing

50 are analysed free — try those first.

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

Integration smooths but does not invert

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