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

The kernel x minus t integrates twice

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

  1. A.f(0) = 0✓ Correct
  2. B.f′(0) exists and f′(0) = 0.✓ Correct
  3. C.f″(0) exists and f″(0) = g(0).✓ Correct
  4. 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, and why this one tests standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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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