I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found [here][1], I need and want to know more about it. Namely

Specifically, I would like to get an integral-residue kind of formulas 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 continuously differentiable on $[0,1]$, then the Euler Maclaurin gives a precise value for $R_f^N$. 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 the analytic functions 
 2. Generalizations to Lebesgue integrals with respect to other measures.
 3. Reminder theorems for continuously differentiable functions.

Most of what I've found were either research papers, or citation of a convenient formula for a specific application. I'm looking for good reviews or textbook chapters which focus on the formula itself.

Thanks

Amir

  [1]: http://www.jstor.org/stable/2589145?seq=1#page_scan_tab_contents