Let $D = f - g$ where $f(x) = \pi \operatorname{cotan}(\pi x)$ and $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and let $\widetilde{D}$ be its continuous extension to $\mathbb{R}$. Justify the existence of $\alpha \in [0,1]$ such that $\widetilde{D}(\alpha) = M$, where $M = \sup_{t \in [0,1]} \widetilde{D}(t)$, then show that:
$$\forall n \in \mathbb{N}, \quad \widetilde{D}\left(\frac{\alpha}{2^n}\right) = M$$