A function $f:[a,b]\to\Bbb R$ is said to be Riemann integrable on $[a,b]$ if there exists a number $L\in\Bbb R$ such that for every $\varepsilon>0$ there exists $\delta_{\varepsilon}>0$ such that if P is any tagged partition of $[a,b]$ with $\Vert$P$\rVert<\delta_{\varepsilon}$ then $|S(f;$P$)-L|<\varepsilon$.
Where $S(f;$P$)$ is the Riemann sum of f corresponding to P.
I am not understanding the definition; I think the part
$|S(f;$P$)-L|<\varepsilon$ is to make sure the function value exists at the tagged value.
But what for the part $\lVert$P$\rVert<\delta_{\varepsilon}$, which is always should be true as there is always a value that is larger than the norm of any partitioned interval...