Riemann Integrability on a Closed Interval

Let $a,b\in\mathbb{R}$ with $a<b$, and let $f:[a,b]\to\mathbb{R}$. The function $f$ is Riemann integrable on $[a,b]$ if there exists a real number $I$ such that for every $\varepsilon>0$ there exists $\delta>0$ with the following property: whenever $P$ is a partition of $[a,b]$ with $|P|<\delta$, and whenever one chooses a tagged partition of $[a,b]$ relative to $P$, the corresponding Riemann sum $S$ of $f$ satisfies $|S-I|<\varepsilon$. In that case $I$ is called the Riemann integral of $f$ over $[a,b]$ and is denoted by $\int_a^b f(x)\,dx.$