# Disjoint intervals for Riemann Integral

So the beginning of Rudin's Real and Complex Analysis states that the Riemann integral on an interval $[a,b]$ can be approximated by sums of the form $\Sigma$$\stackrel{i=1}{n}$f(ti)m(Ei) where the Ei are disjoint intervals whose union is the whole interval.

At least when I learned it, the Riemann integral was partitioned and tags were taken from with the interval [xi, xi+1] which does not form a disjoint collection of intervals.

Can you allow tags to come only from the open interval (xi, xi+1) for some $i$ and from the closed intervals for others, so that we do in fact get a disjoint collection whose union is [a,b]. (this obviously can't work generally for Darboux sums are the inf and sup might not be contained in the image of the interval).