Is the following statement true? |f| Riemann integrable ⇒ f Riemann integrable

No. f = 1 on ℚ ∩ [0,1], −1 elsewhere |f| ≡ 1 is integrable; f is discontinuous everywhere.

Counterexample

Is this true?

|f| Riemann integrable ⇒ f Riemann integrable

No — it is false.

The counterexample

f = 1 on $\mathbb{Q} \cap [0,1], -1$ elsewhere

Why it fails

|f| ≡ 1 is integrable; f is discontinuous everywhere.

Keep practicing

Converse assumed

The theorem runs one way. You used it in the other.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.