Let $M = \mathbb{R}$ and check that the following map is a derivation at $p \in \mathbb{R}$: $$ D(f) = \begin{cases} \lim\limits_{x \to p} \dfrac{f(x) - f(p)}{\;|x-p|^{r + \frac{1}{2}}} &\text{if the limit exists} \\[1ex] 0 &\text{otherwise} \end{cases}. $$
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