I'm using the Euler-Maclaurin formula in a research I'm working on. While brilliant, the elementary proof found here does not give me enough detail.

Specifically, I would like to get an integral-residue kind 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 then that of analytic functions
2. Generalizations to Lebesgue integrals with respect to other measures.
3. Reminder theorems for continuously differentiable functions.

Thanks, Amir