# Riemann Integrability of Step Function

Call $f: [a,b] \to \mathbb{R}$ a step function if there exist 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'm having some issues proving that f is Riemann integrable. If $f$ was defined such that it was constant on the closed interval $[x_i, x_{i+1}]$ then it would be trivial to define a partition such that the supremum and infinum coincide, and the upper and lower sums would cancel. But since we are working with a half-open interval, I'm having issues defining the partitions that will allow me to prove integrability. All proofs I've seen online rely on theorems we haven't proven in class, so I'm assuming I must be missing something simple.

Let $\epsilon>0$ be given, let $M=\max_{x,y\in[a,b]}|f(x)-f(y)|$, and since we have $n+1$ jumps, set $\delta=\frac{\epsilon}{M(n+1)}$. Now consider the partition $P'=\{x_0,x_0+\delta,x_1\pm\delta,x_2\pm\delta,...,x_{n-1}\pm\delta,x_n-\delta,x_n\}$. Can you bound $U(P',f)-L(P',f)$ by something small?