This document introduces a notion of functional differences in which the difference of a function f with respect to a function h is that function g that describes how the value of f changes when its argument is altered by h: f(h(x) = g(f(x)). The author introduces the inverse operation of functional integration and derive useful properties of both operations. The result is a calculus that facilitates derivation and reasoning about recursive programs. This is illustrated in a number of simple examples. The present report uses algebraic methods to establish preliminary results pertaining to fixed differences, that is, functional differences that do not depend on the value of the argument x. Keywords: Theorems; Integrals.