Why a continuous space?
All the examples that we have seen so far in the book had a discrete action space, so you might have the wrong impression that discrete actions dominate the field. This is a very biased view, of course, and just reflects the selection of domains that we picked our test problems from. Besides Atari games and simple, classic RL problems, there are many tasks that require more than just making a selection from a small and discrete set of things to do.
To give you an example, just imagine a simple robot with only one controllable joint that can be rotated in some range of degrees. Usually, to control a physical joint, you have to specify either the desired position or the force applied.
In both cases, you need to make a decision about a continuous value. This value is fundamentally different from a discrete action space, as the set of values on which you can make a decision is potentially infinite. For instance, you could ask the joint to move to a 13.5&...