Question

Which of the following statement is NOT CORRECT?

• A. Every continuous function is integrable
• B. If \(f\) is monotonic on \([a,b]\), then \(f\) is integrable in \([a,b]\)
• C. If \(|f|\) is integrable on \([a,b]\), then \(f\) is integrable on \([a,b]\)
• D. If \(f\) is integrable on \([a,b]\), then \(f^2\) is also integrable on \([a,b]\)

Hint:

If \(f\) is integrable, then \(|f|\) is integrable, but the converse need not be true.

Answer 1

The Correct Option is C

Concept:
There are some standard results in Riemann integration: \[ \text{Continuous function on }[a,b]\Rightarrow \text{integrable} \] \[ \text{Monotonic function on }[a,b]\Rightarrow \text{integrable} \] \[ f\text{ integrable}\Rightarrow |f|\text{ integrable} \] But the converse \[ |f|\text{ integrable}\Rightarrow f\text{ integrable} \] is not always true.

Step 1: Check option (A). Every continuous function on a closed interval is Riemann integrable. So option (A) is correct.

Step 2: Check option (B). Every monotonic function on a closed interval is Riemann integrable. So option (B) is correct.

Step 3: Check option (C). If \(|f|\) is integrable, \(f\) need not be integrable. For example, take \[ f(x)= \begin{cases} 1, & x\in Q, -1, & x\notin Q. \end{cases} \] Then, \[ |f(x)|=1 \] is integrable, but \(f\) is not Riemann integrable. So option (C) is not correct.

Step 4: Check option (D). If \(f\) is Riemann integrable on \([a,b]\), then \(f^2\) is also Riemann integrable. So option (D) is correct.

Step 5: Final answer. \[ \boxed{\text{If }|f|\text{ is integrable on }[a,b],\text{ then }f\text{ is integrable on }[a,b]} \]