What is step function?

Example 15.5 (Piecewise continuous, no drama)

The floor function on $\intcc{0}{3}$ is a step function in disguise: splitting at its jumps,

and the values at the jump points $1, 2$ are irrelevant: changing a function at finitely many points changes no integral (the framing step functions are unaffected). This is the whole content of the “piecewise continuous” extension: cut at the finitely many discontinuities, integrate each continuous piece, add — Chasles as a definition.

Example 15.6 (The definition computes, once)

Let $f(x) = x$ on $\intcc{0}{1}$ and cut into $n$ equal pieces. The best step functions constant on the pieces are $\varphi = \frac{k-1}{n}$ and $\psi = \frac kn$ on the $k$-th piece, with

Every lower integral is $\leq I_-(f) \leq I_+(f) \leq$ every upper one, so $\frac{n-1}{2n} \leq I_-(f) \leq I_+(f) \leq \frac{n+1}{2n}$ for all $n$: both squeeze onto $\frac12$, and $\int_0^1 x\,\dd x = \frac12$ straight from the definition. The closing insight: this is the first and last time we integrate from the definition — the fundamental theorem below replaces all such computations by one antiderivative lookup, which is the entire economic point of this chapter.