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c/d_e$; the latter satisfies the condition ($eq:condimp1$). Let us consider a finite setting, that is, take the universe${\cal U} = {\mathbb{R}}$. Let $$U_0 := \{ v \in {\mathbb{R}}: \sqrt{v^2 + 1} \leq 2\} = \{ v \in {\mathbb{R}}: \sqrt{v^2 + 1} < 2\}$$$U_0$is a bounded open set of${\mathbb{R}}$and it has a non-empty interior (that is, it is an interval). Let$U_1 := \{ v \in {\mathbb{R}}: \sqrt{v^2 + 1} < 2 \} = (-\infty, -1) \cup (1, +\infty)$. Then,$U_0$and$U_1$are two disjoint and clopen set of${\mathbb{R}}$, so, as it is easy to see, they form a$\pi$-system. Let$D_0$be the Dunford–Pettis$\pi$-system generated by$U_0$and$D_1$be the Dunford–Pettis$\pi$-system generated by$U_1$. Then,$D_0 \sqsubseteq D_1 \subseteq U_0$and$D_1\$ satisfies the condition ($eq:condimp1$). [10]{} J. van Benthem. The canonical characterization of infinitesimal space- and time-travel., 62(3):735–740, 2011. J. van Benthem. The [W]{}-theory of infinitesimals., 43(7):1121–1138, 2012. J. van Benthem. On the possibility of construing the calculus of continuous fractions., 138(7):2511–2533, 2012. J. van Benthem, I. Gaspar, and H. Vicar. An infinitesimal calculus which requires no metric., 14(3):275–282, 2005. J. van Benthem

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t i v e o f g ( x ) w r t x ? 2 4 * x * * 2 L e t u ( x ) = 4 * x – 3 . L e t w b e u ( 2 ) . S u p p o s e w – 2 3 = – 4 * s . F i n d t h e s e c o n d d e r i v a t i v e o f – y * * s – y – 2 + 2 w r t y . – 1 2 * y * * 2 L e t c ( j ) b e t h e f i r s t d e r i v a t i v e o f – j * * 5 / 5 – 3 * j * * 2 – 8 . F i n d t h e s e c o n d d e r i v a t i v e o f c ( m ) w r t m . – 1 2 * m * * 2

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