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

Answer: (A), (B), (C)

f is Riemann integrable on [0,1].

Answer 2

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].