We have $f: [a,b] \to \mathbb{R}$ is a step function if there exists a partition $P=\{x_0, \ldots, x_n \}$ of $[a,b]$ such that $f$ is constant on the interval $[x_i, x_{i+1})$.
I want to show that this piecewise constant function is Riemann-integrable. I know by Riemann's Criterion, $f$ is R-integrable $\iff$ $\forall\ \epsilon > 0$, there is a partition $P$ of $[a, b]$ such that $|U(f, P) - L(f, P)| < \epsilon$.
Now it's clear that the thing to tackle here are the discontinuities at the end of each 'piece' of the function. Stuck on how to do that exactly, idea probably is to bound each discontinuity with the partition I choose.
Or maybe an alternative proof could use the additivity of integrals, and define the function on each 'piece'.