Elementary Differential Calculus on Discrete and Hybrid Structures

摘要

We set up differential calculi in the Cartesian-closed category CONV of convergence spaces. The central idea is to uniformly define the 3-place relation __is a differential of__at__ for each pair ofconvergence spaces X, Y in the category, where the first and second arguments are elements of Hom(X, Y) and the third argument is an element of X, in such a way as to (1) obtain the chain rule, (2) have the relation be in agreement with standard definitions from real and complex analysis, and (3) depend only on the convergence structures native to the spaces X and Y. All topological spaces and all reflexive directed graphs (I.e. discrete structures) are included in CONV. Accordingly, ramified hybridizations of discrete and continuous spaces occur in CONV. Moreover, the convergence structure within each space local to each point, individually, can be discrete, continuous, or hybrid.