Part CCSIR NET December 2024the-kernel-x-minus-t-integrates-twice

The kernel x minus t integrates twice

Let $g : R \to R$ be a continuous function. Define $f (x) = \int_{0}$ ˣ(x − t)g(t)dt, $x \in R$ . Which of the following statements are true?

A.f(0) = 0✓
B.f′(0) exists and f′(0) = 0.✓
C.f″(0) exists and f″(0) = g(0).✓
D.f″(0) exists but f″(0) ≠ g(0).

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.

50 are analysed free — try those first.

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

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

From Integration › Riemann integration and criteria

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