一句话：用邻近格上的可逼近映射与 Stone 对偶，为局部 Lipschitz 映射的 Clarke 次梯度建立逻辑形式的微分理论。
本文为有限维欧氏空间上的实值函数提出一种逻辑形式的微分理论。核心问题是如何用 domain/logical 结构刻画连续函数、局部 Lipschitz 映射及其 Clarke 次梯度（等价于 L-导数）。方法上，用半强邻近格之间的局部可逼近映射表示连续函数，用可逼近映射表示非空紧凸值的 Clarke 次梯度；这类映射构成有界完全 domain，并与相应 Scott 连续函数空间同构。作者引入 single-knot、strong single-tie 与 strong knot，建立局部 Lipschitz 映射和 Lipschitzian approximable mappings 的 Stone 对偶，并用 strong single-knot 定义 Lipschitzian 导数。最后给出若干基本构造子的导数演算，证明其与 Clarke 次梯度演算对偶。
We introduce a logical theory of differentiation for a real-valued function on a finite dimensional real Euclidean space. A real-valued continuous function is represented by a localic approximable mapping between two semi-strong proximity lattices, representing the two stably locally compact Euclidean spaces for the domain and the range of the function. Similarly, the Clarke subgradient, equivalently the L-derivative, of a locally Lipschitz map, which is non-empty, compact and convex valued, is represented by an approximable mapping. Approximable mappings of the latter type form a bounded complete domain isomorphic with the function space of Scott continuous functions of a real variable into the domain of non-empty compact and convex subsets of the finite dimensional Euclidean space partially ordered with reverse inclusion. Corresponding to the notion of a single-tie of a locally Lipschitz function, used to derive the domain-theoretic L-derivative of the function, we introduce the dual notion of a single-knot of approximable mappings which gives rise to Lipschitzian approximable mappings. We then develop the notion of a strong single-tie and that of a strong knot leading to a Stone duality result for locally Lipschitz maps and Lipschitzian approximable mappings. The strong single-knots, in which a Lipschitzian approximable mapping belongs, are employed to define the Lipschitzian derivative of the approximable mapping. The latter is dual to the Clarke subgradient of the corresponding locally Lipschitz map defined domain-theoretically using strong single-ties. A stricter notion of strong single-knots is subsequently developed which captures approximable mappings of continuously differentiable maps providing a gradient Stone duality for these maps. Finally, we derive a calculus for Lipschitzian derivative of approximable mapping for some basic constructors and show that it is dual to the calculus satisfied by the Clarke subgradient.