I'm using the Euler–Maclaurin formula in a research project I'm working on. While brilliant, the elementary proof found in [Apostol - An Elementary View of Euler's Summation Formula](https://doi.org/10.2307/2589145) does not give me enough detail.


Specifically, I would like to get an integral-residue kind of formula for functions which are continuously differentiable only on open intervals. **To be precise**:


Consider $f:(0,1) \to \mathbb{R}$ a continously differentiable function, and define $$R^N\_f := \sum\limits\_{m=1}^{N} f\left( \frac{m}{N} \right) - \int\limits\_0^1 f(t)~ dt ~ . $$


If $f$ is also continuously differentiable on $[0,1]$, then the Euler–Maclaurin formula gives a closed-form expression for $R\_f^N$ (which depends on an unknown point). If the integral on the RHS exists, but the function is not continuously differentiable on the closed interval — what can be said about the error term?


**More generally speaking**, if no such result exists, I'm interested in


1. Generalizations to broader function spaces than that of analytic functions.
2. Generalizations to Lebesgue integrals with respect to other measures.
3. Reminder theorems for continuously differentiable functions.



Generalizations of the Euler–Maclaurin Summation Formula
As for point (1), maybe the following references will be useful:


* [The Euler–Maclaurin formula revisited, by D. Elliott](https://www.researchgate.net/publication/228707872_The_Euler-Maclaurin_formula_revisited)
* [The Euler–Maclaurin expansion and finite-part integrals, by G. Monegato, J.N. Lyness](https://doi.org/10.1007/s002110050392) and references in it, especially [8] (I. Navot) and [9] (B.W. Ninham).


