Learning from demonstration (LfD) provides an efficient way to train robots. The learned motions should be convergent and stable, but to be truly effective in the real world, LfD-capable robots should also be able to remember multiple motion skills. Multi-skill retention is a capability missing from existing stable-LfD approaches. On the other hand, recent work on continual-LfD has shown that hypernetwork-generated neural ordinary differential equation solvers, can learn multiple LfD tasks sequentially, but this approach lacks stability guarantees. We propose an approach for stable continual-LfD in which a hypernetwork generates two networks: a trajectory learning dynamics model, and a trajectory stabilizing Lyapunov function. The introduction of stability not only generates stable trajectories but also greatly improves continual learning performance, especially in the size-efficient chunked hypernetworks. With our approach, we can continually train a single model to predict the position and orientation trajectories of the robot's end-effector simultaneously for multiple real world tasks without retraining on past demonstrations. We also propose stochastic regularization with a single randomly sampled regularization term in hypernetworks, which reduces the cumulative training time cost for $N$ tasks from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ without any loss in performance in real-world tasks. We empirically evaluate our approach on the popular LASA dataset, on high-dimensional extensions of LASA (including up to 32 dimensions) to assess scalability, and on a novel extended robotic task dataset (RoboTasks9) to assess real-world performance. In trajectory error metrics, stability metrics and continual learning metrics our approach performs favorably, compared to other baselines. Code and datasets will be shared after submission.