real analysis - Some questions about Riemann integration

I just learn some basic definition about

Let $f, g:[0,1] \rightarrow \Bbb R$ be

$ f(x) =
\begin{cases}
1, & \text{if $x \in \Bbb Q$} \\
0, & \text{otherwise}
\end{cases}$

$g(x) =
\begin{cases}
1, & \text{if $x=1/n, n=1,2,...$} \\

0, & \text{otherwise}
\end{cases}$

We know $f$ is not Riemann integrable, but $g$ is.

So my first question is, is it true that if the set of discontinuous points is a dense set, then that function is not Riemann integrable.

My second question is we know $h:[0,1] \rightarrow \Bbb R$ by $h(x)=1$ is integrable and has value $1$. So if we have a dense set $D$ in $[0,1]$ which cardinality of $D$ and $D^c$ are equal, and define $ u(x) =
\begin{cases}
1, & \text{if $x \in D$} \\

0, & \text{otherwise}
\end{cases}$

Can we define a similar 'integral' to say the value of the 'integral' = $1/2$

Thank you!