They are valued because they are countable and transferable, and selfreproducing stores of milk, which like money is slippery.
One suspects that there is indeed something interesting about an object of study that is deemed countable and delimitable yet defies both activities.
It follows that f has left and right limits at each point, and that f is continuous except at a countable set of points.
In our more general topological setting, a natural first attempt at doing something similar is to restrict to compactly generated spaces with countable (topological) bases.
In spite of the countable, dense set of periodic orbits, the dynamics of f are a prototype of chaotic behaviour.
For example, one may be able to understand amenability for general locally compact groupoids by reducing some questions to the case of countable orbits.
In the same way, some events are countable and atomic (thus having dog-like properties), whereas others are countable and cumulative (thus having fence-like properties).
However, for the purpose of studying finite or countably infinite automata, it suffices to work with the following notion of a (countable) probability presentation.
