We introduce the Continuous Edit Distance (CED), a geodesic and elastic distance for time-varying persistence diagrams (TVPDs). The CED extends edit-distance ideas to TVPDs by combining local substitution costs with penalized deletions/insertions, controlled by two parameters: \(α\) (trade-off between temporal misalignment and diagram discrepancy) and \(β\) (gap penalty). We also provide an explicit construction of CED-geodesics. Building on these ingredients, we present two practical barycenter solvers, one stochastic and one greedy, that monotonically decrease the CED Frechet energy. Empirically, the CED is robust to additive perturbations (both temporal and spatial), recovers temporal shifts, and supports temporal pattern search. On real-life datasets, the CED achieves clustering performance comparable or better than standard elastic dissimilarities, while our clustering based on CED-barycenters yields superior classification results. Overall, the CED equips TVPD analysis with a principled distance, interpretable geodesics, and practical barycenters, enabling alignment, comparison, averaging, and clustering directly in the space of TVPDs. A C++ implementation is provided for reproducibility at the following address https://github.com/sebastien-tchitchek/ContinuousEditDistance.