When you $\tt{diff}$erence discrete observations of a function $f$ you lose one observation each time you apply $\tt{diff}$.
When you $\partial$ifferentiate a $\mathcal{C}^n \ni$ function $f: \mathbb{R} \to \mathbb{R}$, losing an observation would make the cardinality of the domain $\aleph_1-1$ which does nothing since $\aleph_1-1 = \aleph_1$. So we require $f' \in \mathcal{C}^{n-1}$ instead. In an epsilon-delta sense this is like saying $$\forall \varepsilon > 0, \ \exists \delta \text{ such that } \lim_{h \downarrow 0} { f((x + \delta)-h) - f(x + \delta) \over h } < \lim_{h \downarrow 0} {f(x+h) - f(x) \over h} \pm \varepsilon.$$
I've also heard the saying "Calculus is topology" because of the similarity of the boundary operator $\partial$ to the derivative operator $\partial$, as could be seen in Green's theorem.
Is there a deeper connection between "lose one observation" and "lose a continuity"? If so, what are the next steps to understand it?