While studying Analysis, I used many different textbooks: "Understanding Analysis" (Stephen Abbott), "Elements of Real Analysis" (Denlinger), "Calculus" (Michael Spivak).
Initially when dealing with integrals in Calculus, we use the Riemann Integral as a limit of equally spaced partitions (some people may call it standard partitions), here is a picture of a common way of defining it: picture from "Calculus Vol.1" (James Stewart).
- For this question, please ignore the fact that in this image the author restricts the integral to the use of the left point for each interval, focusing only on the regular partitions problem.
However, when dealing with a more profound subject like Analysis, I realized that there are many different ways to define the integral, relating to the Riemann Integral there are two main procedures: the Darboux Approach and the Riemann Sum Approach. Both of them are equivalent, that is, a function is Darboux Integrable if and only if it is Riemann Integrable (it is also common to call both of the integrals the Riemann Integral since they are equivalent, as far as I know), as it is proved in many books (see, for example, pg. 215, theorem 8.1.2 of "Understanding Analysis" (Stephen Abbott)).
However, there is one thing I'd like to know: is the equivalence of the Darboux and Riemann Sum approaches preserved when we use definitions such as picture from "Calculus Vol.1" (James Stewart) or picture from "Elements of Real Analysis (Denlinger)"? That is, when integrating with these methods of equally spaced partitions, will it still be equivalent to the Darboux Approach and the non-standard partitions approaches for the Riemann Sum?