Weierstrass theorem says that any continuous function on a compact interval is bounded. Is this also true for functions continuous except countably many points (ex. monotone functions)? If not, how monotone functions (so continuous except countably many points) on compact intervals are always Riemann integrable?
Every monotone function on $[a,b]$ is bounded, because $f(x)$ is always between $f(a)$ and $f(b)$; hence $|f(x)|\le \max(|f(a)|,|f(b)|)$.
Boundedness is not enough for Riemann integrability, but monotonicity helps again: the same in-betweenness idea applies on subintervals, as in this answer by Robert Israel.
Without monotonicity, even one discontinuity point can make a function nonintegrable, like $f(x)=1/x^2$ on $[-1,1]$.