Question

Define f: [0,1] → [0,1] by
\(f(x) = \begin{cases} 1 & \text{if } x =0 \\ \frac{1}{n} & \text{if } x=\frac{m}{n}\ \text{for some}\ m, n \isin \N\ \text{with}\ m\le n\ \text{and gcd(m, n)} = 1, \\ 0 & \text{if} x\isin[0,1]\ \text{is irrational}\end{cases}\)
and define g: [0,1]→ [0,1] by
\(g(n) = \begin{cases} 0 & \text{if } x=0 \\ 1 & \text{if } x\isin(0,1]. \end{cases}\)
Then which of the following is/are true?

• A. f is Riemann integrable on [0,1].
• B. g is Riemann integrable on [0,1].
• C. The composite function f○g is Riemann integrable on [0,1].
• D. The composite function g○f is Riemann integrable on [0,1].

Answer 1

The Correct Option is A, B, C

The correct option is (A): f is Riemann integrable on [0,1]., (B): g is Riemann integrable on [0,1]. and (C): The composite function f○g is Riemann integrable on [0,1].