Rapid growth in speech data demands adaptive models, as traditional static methods fail to keep pace with dynamic and diverse speech information. We introduce continuous speech learning, a new set-up targeting at bridging the adaptation gap in current speech models. We use the encoder-decoder Whisper model to standardize speech tasks into a generative format. We integrate a learnable gated-fusion layer on the top of the encoder to dynamically select task-specific features for downstream tasks. Our approach improves accuracy significantly over traditional methods in six speech processing tasks, demonstrating gains in adapting to new speech tasks without full retraining.