In Rene Schilling - Measures, Integrals and Martingales, on page 94, Theorem 11.8 states that:
A bounded function $f:[a,b]\rightarrow\mathbb{R}$ is Riemann integrable if, and only if, the points in $(a,b)$ where f is discontinuous are a Lebesgue null set.
Well, then I don't really understand why the function $f(x)=\frac{1}{x}$ is not Riemann integrable on $[0,1]$ (of course I actually do, cause it is not bounded there) but $g(x)=\ln{x}$ is.
Why is happening this? $g$ is unbounded in the same interval but still Riemann integrable (it's integral is -1)
Which differs from $f$ to $g$?