In Part 1 of this paper (Function Data Types) we developed ultra-arithmetic. a calculus for functions which is performable on a digital computer. Here in analogy with the notion of interval arithmetic for intervals of reals we begin the development of an interval arithmetic for functions. The operations of addition, subtraction, multiplication, division, integration and differentiation for intervals of polynomials are defined and studied. In certain cases simplified isotonal approximations to the resultant interval as well as error analyses are also given. © 1982.