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Conventional text-to-SQL studies are limited to a single task with a fixed-size training and test set. When confronted with a stream of tasks common in real-world applications, existing methods struggle with the problems of insufficient supervised data and high retraining costs. The former tends to cause overfitting on unseen databases for the new task, while the latter makes a full review of instances from past tasks impractical for the model, resulting in forgetting of learned SQL structures and database schemas. To address the problems, this paper proposes integrating semi-supervised learning (SSL) and continual learning (CL) in a stream of text-to-SQL tasks and offers two promising solutions in turn. The first solution Vanilla is to perform self-training, augmenting the supervised training data with predicted pseudo-labeled instances of the current task, while replacing the full volume retraining with episodic memory replay to balance the training efficiency with the performance of previous tasks. The improved solution SFNet takes advantage of the intrinsic connection between CL and SSL. It uses in-memory past information to help current SSL, while adding high-quality pseudo instances in memory to improve future replay. The experiments on two datasets shows that SFNet outperforms the widely-used SSL-only and CL-only baselines on multiple metrics.