Can regression trees predict continuously?
Regression trees, particularly gradient boosting (essentially many trees), tend to do very well on continuous predictions, often outperforming models that are truly continuous like linear regression when. This is especially true when there are variable interactions and when you have a sufficiently large dataset (over 10,000 records) so that overfitting is less likely. If your primary objective is simply predictive power, then whether the model is 100% continuous or pseudo continuous should be irrelevant. If making your regression trees more continuous enhances out of sample predictive power, then you can simply increase tree depth or add more trees.