Give an example of a function $f:[0,1] \rightarrow \mathbb{R}$ such that...
(a) $f$ is bounded, but not Riemann integrable on $[0,1]$. $ f(x) := \begin{cases} 2x & \text{if $x$ is rational}\\ x & \text{if $x$ is irrational.} \end{cases} $ (b) $f$ is Riemann integrable on $[0,1]$ but not monotone.
$f(x) := 2$
(c) $f$ is Riemann integrable on $[0,1]$ but neither continuous nor monotone.
$f(x) := \begin{cases} 0 & \text{if $x$ is $0$}\\ 2 & \text{otherwise.} \end{cases} $
Is this correct? Thanks!