What exactly is a distinguished point in the Riemann sum when defining the definite integral 
I do not understand what a distinguished point is. Is it a real number between endpoints $x_{i-1}$ and $x_{i}$, or something else?
(source https://en.wikipedia.org/wiki/Integral#Formalization)
 A: There are various constructions of the definite Riemann integral $\int_a^b f(x)\>dx$. All of them work with partitions
$${\cal P}:\quad a=x_0<x_1<x_2<\ldots<x_N=b$$
of the interval $[a,b]$, and some of them work with tagged partitions $({\cal P}, {\bf t})$  (or similarly notated), where in addition to the separating points $x_i$ of ${\cal P}$ we choose in each interval $[x_{i-1},x_i]$ $\>(1\leq i\leq N)$  an "evaluating point" $t_i$. These points $t_i$ are in no way "distinguished". The $t_i$ are just $N$ more "dummy variables" entering in the limiting process.
The result of this limiting process is always the same, and is denoted by $\int_a^b f(x)\>dx$. When we just have partitions ${\cal P}$ we need additional "local infs and sups" to form lower and upper Riemann sums (i.e., "hidden limits")
$$L(f,{\cal P}):=\sum_{i=1}^N \inf\bigl\{f(t)\bigm|x_{i-1}\leq t\leq x_i\bigr\}\>(x_i-x_{i-1}),\qquad U(f,{\cal P}):=\ldots\ .$$
With tagged partitions we just look at sums
$$R(f,{\cal P}):=\sum_{i=1}^N f(t_i)\>(x_i-x_{i-1})\ ,$$
whereby the $t_i$-data are tacitly assumed, but not referred to in the $R$-notation.
A: If you remember the left-hand sum, right-hand sum, trapezoidal sum, etc. Those are ways of choosing your distinguished points and partitions. The integral exists when you are allowed to choose any distinguished point for any partition and take the limit as the length of your partitions goes to zero.


