Index sets for computable differential equations

dx = F (x, y) and the wave equation from the point of view of index sets. Index sets play an important role in the study of computable functions and computably enumerable sets (see, for example, Soare [21]) . Index sets for computable combinatorics have been studied by Gasarch and others [8, 9]; the latter paper provides a survey of such results. Index sets for Π1 classes were developed by the authors in [3] and applied to several areas of computable mathematics including computable algebra and logic, computable orderings, computable combinatorics, and computable analysis. In this paper, we use index sets to develop a complexity measure for the class of computably continuous functions. This follows the path of four recent papers [3, 5, 6, 7] where we studied index sets for Π1 classes and computably continuous functions. The results of those papers assign a precise level of complexity in the arithmetic hierarchy to various properties of classes and functions. For example, the complexity of a set having measure one is Π1 complete, the complexity of a set having cardinality ≥ 2 is Σ2 complete and the complexity of a function having a computable fixed point is Σ3 complete. The key to the development of a successful theory of index sets for various properties associated with the derivatives of computably continuous functions is to choose an appropriate definition of an index of a computably continuous function. We define the notion of an index for a computable real function of n variables by defining a Π2 set I n of indices a such that the computable function φa defines a computable real function Fa : R −→ R , where R denotes the reals. In fact, I is Π2 complete. This means that the most meaningful index set results that we obtain involve conditions whose complexity is greater than Π2. Nevertheless, there are a number of results that we can obtain for less complex conditions. For example, we show that {〈a, b〉 ∈ I × I : ∂Fa ∂xi = Fb} is a