Let $f$ be a real function with $\lim_{n\rightarrow\infty}f(x)=0$.
(1) What is the necessary conditions for the Mueller's formula:
$$ \sum_{x}f(x)=\sum_{n=1}^\infty(f(n)-f(n+x))+C ? $$ (2) What is $C$?
(3) Any references about it, Euler–Maclaurin formula and Antidifferences of special functions (e.g., eponential, logarithm, trigonometric, etc.)?