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?