Ordinary differential equations (ODEs) are often used to model the dynamics of (often safety-critical) continuous systems. This work presents the formal verification of an algorithm for reachability analysis in continuous systems. The algorithm features adaptive RungeKutta methods and rigorous numerics based on affine arithmetic. It is proved to be sound with respect to the existing formalization of ODEs in Isabelle/HOL. Optimizations like splitting, intersecting and collecting reachable sets are necessary to analyze chaotic systems. Experiments demonstrate the practical usability of our developments.