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Existing domain adaptation systems can hardly be applied to real-world problems with new classes presenting at deployment time, especially regarding source-free scenarios where multiple source domains do not share the label space despite being given a few labeled target data. To address this, we define a novel problem setting: multi-source semi-supervised open-set domain adaptation and propose a learning theory via joint error, effectively tackling strong domain shift. To generalize the algorithm into source-free cases, we introdcue a computationally efficient and architecture-flexible attention-based feature generation module. Extensive experiments on various data sets demonstrate the significant improvement of our proposed algorithm over baselines.