The goal of rigorous function calculus is to work with functions on continuous spaces (notably Euclidean space) in a way which is both mathematically rigorous and computationally efficient. I will start by giving types for the main classes of function, notably continuous, smooth, measurable, piecewise and multivalued functions, and introduce the relation between effective and validated information. I will then give details of the implementation of rigorous function calculus in the tool Ariadne, including concrete polynomial function models and their use for the analysis and verification of dynamic systems. Finally, I will talk about some open problems, which hopefully we can work on during my stay in Kyoto.