Smooth infinitesimal analysis

Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete entities. As a theory, it is a subset of synthetic differential geometry. Terence Tao has referred to this concept under the name "cheap nonstandard analysis."

The nilsquare or nilpotent infinitesimals are numbers ε where ε² = 0 is true, but ε = 0 need not be true at the same time. Calculus Made Easy notably uses nilpotent infinitesimals.