Deep learning models achieve high accuracy in signal processing but remain difficult to interpret. Classical matching pursuit and sparse coding offer better transparency but suffer from spectral leakage and grid mismatch when signal frequencies do not align with discrete dictionary atoms. We introduce Continuous Dictionary Pursuit (CDP), a framework for interpretable signal decomposition that learns the parameters of closed-form functions directly from data. CDP optimizes over a continuous manifold of differentiable atoms and, unlike prior continuous pursuit methods that operate on a single analytic family, searches a heterogeneous library spanning periodic, trend, transient, and discontinuous structures. The algorithm employs a greedy iterative strategy using data-driven priors and gradient-based optimization to isolate individual signal components, with discontinuous waveforms made differentiable through spectral sharpness annealing. To ensure a sparse atom set for reconstruction, the framework utilizes a stopping criterion based on a predefined atom budget and residual error convergence. Evaluations on synthetic and real-world datasets show that CDP removes the grid discretization error for atom families contained in its library, recovers generative parameters to within
1% on stationary synthetic signals expressible in that library, and achieves the best sparsity-accuracy trade-off among classical, continuous-pursuit, and neural baselines in the sparse regime. Beyond decomposition, we demonstrate that the extracted atoms can serve as structured curriculum targets for training sequence models, improving forecasting accuracy over standard end-to-end training on most evaluated datasets. The method offers a glass-box alternative for signal decomposition by providing high reconstruction fidelity alongside direct physical interpretability.