Question

Consider the following statements for a function \(f:[a,b]\to\mathbb{R}\):
(I) \(f\) is Riemann integrable if and only if it is bounded on \([a,b]\).
(II) \(f\) is Riemann integrable if and only if it is monotonic on \([a,b]\).
(III) \(f\) is Riemann integrable if and only if it is continuous on \([a,b]\).
Which one of the following options is correct?

• A. Only (III) is true
• B. Only (I) and (II) are true
• C. Only (II) and (III) are true
• D. All (I), (II) and (III) are false

Hint:

Bounded, monotonic and continuous are each only sufficient for integrability, never necessary.

Answer 1

The Correct Option is D

Step 1: Recall the exact criterion for Riemann integrability. A bounded function \(f\) on \([a,b]\) is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero (Lebesgue's criterion). Boundedness alone, monotonicity alone, and continuity alone are each only sufficient conditions, not equivalent characterisations.

Step 2: Check statement (I). Integrability implies boundedness, but boundedness does not imply integrability. The Dirichlet function \(f(x)=1\) for rational \(x\) and \(f(x)=0\) for irrational \(x\) on \([0,1]\) is bounded but discontinuous everywhere, so it is not Riemann integrable. Hence (I) is false.

Step 3: Check statement (II). Every monotonic function on \([a,b]\) is Riemann integrable, but the converse fails. The function \(f(x)=x(1-x)\) on \([0,1]\) is continuous, increases then decreases, so it is not monotonic, yet it is Riemann integrable. Hence (II) is false.

Step 4: Check statement (III). Every continuous function on \([a,b]\) is Riemann integrable, but the converse fails. A step function with a finite jump discontinuity is Riemann integrable without being continuous. Hence (III) is false.

Step 5: Since each 'if and only if' claim fails, all three statements (I), (II) and (III) are false.

\[\boxed{\text{All (I), (II) and (III) are false}}\]