Riemann integrable function

Definition

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be a function. The function $f$ is Riemann integrable on $[a, b]$ if there is a real number $I$ with the following property. For each $\varepsilon > 0$ there is $\delta > 0$ such that $|S(f, P, t) - I| < \varepsilon$ for each tagged partition $(P, t)$ of $[a, b]$ with mesh $\| P \| < \delta$. Here $S(f, P, t)$ is the Riemann sum of $f$ for $(P, t)$.

Remarks

The number $I$ is unique. It is the Riemann integral of $f$. A Riemann integrable function is bounded. Continuous functions and monotone functions on $[a, b]$ are Riemann integrable. The Dirichlet function is not.

Explanations

L1 human

A function is Riemann integrable if its Riemann sums settle down to one number as the partitions get finer. It does not matter which tags we choose in the pieces.