Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by:
$$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$
We set $D = f - g$.
Show that the function $D$ extends by continuity to a function $\widetilde{D}$ on $\mathbb{R}$ such that $\widetilde{D}(0) = 0$.