Let $f , g : [ 0,1 ] \rightarrow \mathbb { R }$ be given by $$\begin{gathered}
f ( x ) : = \begin{cases} x ^ { 2 } & \text { if } x \text { is rational, } \\
0 & \text { if } x \text { is irrational; } \end{cases} \\
g ( x ) : = \begin{cases} 1 / q & \text { if } x = \frac { p } { q } \text { is rational, with } \operatorname { gcd } ( p , q ) = 1 , \\
0 & \text { if } x \text { is irrational. } \end{cases}
\end{gathered}$$ Then, (a) $g$ is Riemann integrable, but not $f$; (b) both $f$ and $g$ are Riemann integrable; (c) the Riemann integral $\int _ { 0 } ^ { 1 } f ( x ) d x = 0$; (d) the Riemann integral $\int _ { 0 } ^ { 1 } g ( x ) d x = 0$.
Let $f , g : [ 0,1 ] \rightarrow \mathbb { R }$ be given by
$$\begin{gathered}
f ( x ) : = \begin{cases} x ^ { 2 } & \text { if } x \text { is rational, } \\
0 & \text { if } x \text { is irrational; } \end{cases} \\
g ( x ) : = \begin{cases} 1 / q & \text { if } x = \frac { p } { q } \text { is rational, with } \operatorname { gcd } ( p , q ) = 1 , \\
0 & \text { if } x \text { is irrational. } \end{cases}
\end{gathered}$$
Then,\\
(a) $g$ is Riemann integrable, but not $f$;\\
(b) both $f$ and $g$ are Riemann integrable;\\
(c) the Riemann integral $\int _ { 0 } ^ { 1 } f ( x ) d x = 0$;\\
(d) the Riemann integral $\int _ { 0 } ^ { 1 } g ( x ) d x = 0$.