Continuous attractors offer a unique class of solutions for storing continuous-values in recurrent system states indefinitely.Unfortunately, the continuous attractors suffer from severe structural instability in general---they are destroyed by most infinitesimal changes of the dynamical law.This fragility limits their utility especially in biological systems as their recurrent dynamics are subject to constant perturbations.We observe that the bifurcations from continuous attractors in theoretical neuroscience models display various structurally stable forms.Although their asymptotic behaviors of memory are categorically distinct, their finite time behaviors are similar and degrade gracefully.Fast-slow decomposition analysis uncovers the persistent manifold that survives the seemingly destructive bifurcation.Moreover, the recurrent neural networks trained on analog memory tasks display approximate continuous attractors with predicted slow manifold structures.Therefore, continuous attractors are functionally robust and remain useful as a universal analogy for understanding analog memory.