Part CCSIR NET June 2025the-continuous-function-must-be-the-outer-one-not-the-inner-one

The continuous function must be the outer one not the inner one

Let f and g be real-valued Riemann integrable functions on [a, b] such that g([a, b]) ⊆ [a, b]. Which of the following statements are necessarily true?

A.The composition f ∘ g is Riemann integrable.
B.If g(x) ≠ 0 for each x ∈ [a, b], then f/g is Riemann integrable.
C.The positive square root $f^{2} + g^{2}$ is Riemann integrable.✓
D.The composition f ∘ g is Riemann integrable, if both f and g are continuous.✓

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.

See pricing

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

The continuous function must be the outer one not the inner one

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