We create a computationally tractable algorithm for contextual bandit learning with one-dimensional continuous actions with unknown structure on the loss functions. In a nutshell, our algorithm, Continuous Action Tree with Smoothing (CATS), reduces continuous-action contextual bandit learning to cost-sensitive extreme multiclass classification, where each class corresponds to a discretized action. We show that CATS admits an online implementation that has low training and test time complexities per example, and enjoys statistical consistency guarantees under certain realizability assumptions. We also verify the efficiency and efficacy of CATS through large-scale experiments.