  We introduce a generalization of the Cantor-Dedekind continuum with explicit
infinitesimals. These infinitesimals are used as numbers obeying the same basic
rules as the other elements of the generalized continuum, in accordance with
Leibniz's original intuition, but with an important difference: their product
is null, as the Dutch theologian Bernard Nieuwentijt sustained, against
Leibniz's opinion. The starting-point is the concept of shadow, and from it we
define indiscernibility (the central concept) and monad. Monads of points have
a global-local nature, because in spite of being infinite-dimensional real
affine spaces with the same cardinal as the whole generalized continuum, they
are closed intervals with length 0. Monads and shadows (initially defined for
points) are then extended to any subset of the new continuum, and their study
reveals interesting results of preservation in the areas of set theory and
topology. All these concepts do not depend on a definition of limit in the new
continuum; yet using them we obtain the basic results of the differential
calculus. Finally, we give two examples illustrating how the global-local
nature of the monad of a real number can be applied to the differential
treatment of certain singularities.
