Why are functions with a finite number of (removable or jump) discontinuities Riemann-integrable while Dirichlet function isn't?
(Edit)
More precisely:
$\Bbb Q \,\cap [0,1]=:(a_k)_{k\in \Bbb N}$
Define $f_n(x)$ as follows: $$\begin{cases} 1 \quad \text{if} \,\,x\in \{a_1,\dots,a_n\} \\ 0 \quad \text{otherwise} \end{cases} $$ Why are these functions $f_n$ Riemann-integrable?
I don't know anything about Lebesgue integration yet.