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Part CCSIR NET December 2024monotone-means-countably-many-jumps

Monotone means countably many jumps

Let f:[0,1]→Rf : [0, 1] \to \mathbb{R} be a monotonic function. Which of the following statements are true?

  1. A.f is Riemann integrable on [0, 1].✓ Correct
  2. B.The set of discontinuities of f cannot contain a non-empty open set.✓ Correct
  3. C.f is Lebesgue measurable function.✓ Correct
  4. D.f is Borel measurable function.✓ Correct

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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