I am supposed to find an increasing function on the interval $\left[0, 1\right]$ that is discontinuous for $x \in \{1 / n\}_{n = 2}^{\infty}$ and then explain directly why it is Riemann-Integrable.
I know that a function is Riemann-Integrable if $f$ is bounded on $\left[a, b\right]$ and $I_* = I^*$, where $I_*$ is the least upper bound of the set of values $L(f, P)$ for various partitions P and $I^*$ is the greatest lower bound for the set of values of $U(f, P)$ for various partitions P.
Wondered if someone had a simple example of this to show me? My professor said there is a lot of functions to choose from, but the way to explain it would be the same for all of them. The definition that I wrote here makes sense to me, can't really refer to much else.