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 $(P, t)$.