state gov RPSC 2nd Grade Senior Teacher Mock Test Series 2025 Mathematical Science Analysis Riemann Sums and Riemann Integral
Let\( f: [0, 1] \to \mathbb{R} \) be a bounded function such that for any partition \( P = \{ x_0, x_1, \dots, x_n \} \) of the interval [0, 1], the following condition holds:
\(\sum_{i=1}^{n} \left( \sup_{x \in [x_{i-1}, x_i]} f(x) - \inf_{x \in [x_{i-1}, x_i]} f(x) \right) \to 0 \quad {as} \quad \|P\| \to 0, \)
where \( \|P\| \) is the mesh of the partition.
Which of the following statements is true?
1
f is Riemann integrable on [0, 1], and the integral \(\int_0^1 f(x) \ \) , dx exists and is finite.
2
f is not Riemann integrable on [0, 1] because the condition above is not sufficient for Riemann integrability.
3
f is Riemann integrable on [0, 1] only if the function is continuous almost everywhere.
4
The given condition implies that f is Darboux integrable but not Riemann integrable.
5
Question Not Attempted