TITLE: Do regular partitions suffice for Riemann integrability of a real-valued function on a closed interval?
QUESTION [8 upvotes]: Does the following condition for a bounded function $f: [a, b] \to \mathbb R$ suffice that it be Riemann integrable on $[a,b]$, with Riemann integral $I$?
For every $\epsilon > 0$, there exists a positive integer $N$ such that, for every positive integer $n \geq N$:  for each $i = 1, 2, \dots, n$, for each number $c_{i} \in [x_{i-1}, x_{i}]$ — the $i$th subinterval in the $n$th regular partition of $[a, b]$ whose subintervals have length $\Delta x = (b-a)/n$ — we have
$\left| \sum_{i=1}^{n} f\bigl(c_{i})\,\Delta x - I \right| < \epsilon$.
The condition may be stated equivalently as:
For every $\epsilon > 0$, there exists a $\delta > 0$ such that, for every positive integer $n$ with $\Delta x = (b - a)/n < \delta$:  for each $i = 1, 2, \dots, n$, for each number $c_{i} \in [x_{i-1}, x_{i}]$ — the $i$th subinterval in the $n$th regular partition of $[a, b]$ — we have
$\left| \sum_{i=1}^{n} f\bigl(c_{i})\,\Delta x - I \right| < \epsilon$.
In short, do regular partitions suffice for Riemann integrability? 
Notes:

The usual condition for Riemann integrability involves arbitrary partitions, not just those that are regular, and then of course it involves the mesh of partitions being $< \delta$. Of course for a regular partition with $n$ subintervals, that mesh is $\Delta x = (b-a)/n$.
The only thing I intend to change in the usual condition is to restrict partitions just to those that are regular.
Note that the condition as stated allows the sample points to be chosen arbitrarily in the subintervals. It's well known that it is not enough to allow just endpoints.

REPLY [8 votes]: Regular partitions are enough !
The problem is not banal: see

Charles G. Denlinger Elements of Real Analysis (2011), p. 378 ;
Jingcheng Ton Partitions of the interval in the definition of Riemann's integral Journal of Math. Educ. in Sc. and Tech. 32 (2001), pp. 788-793 (theorem 3).

It is sufficient to use any fixed sequence of partitions with the norm converging to zero (the fact is not much known).