We exploit the temporal coherence of closed-form animated implicit surfaces by locally re-evaluating an octree-like discretization of the implicit field only as and where is necessary to rigorously maintain a global error invariant over time, thereby saving resources in static or slowly-evolving areas far from the motion where per-frame updates are not necessary. We treat implicit surface rendering as a special case of the continuous constraint satisfaction problem of set inversion, which seeks preimages of arbitrary sets under vector-valued functions. From this perspective, we formalize a temporally-coherent set inversion algorithm that localizes changes in the field by range-bounding its time derivatives using interval arithmetic. We implement our algorithm on the GPU using persistent thread scheduling and apply it to the scalar case of implicit surface and swept volume rendering where we achieve significant speedups in complex scenes with localized deformations like those found in games and modelling applications where interactivity is required and bounded-error approximation is acceptable.