  The existence of a Control Barrier Function (CBF) for a control-affine system
provides a powerful design tool to ensure safety. Any controller that satisfies
the CBF condition and ensures that the trajectories of the closed-loop system
are well defined makes the zero superlevel set forward invariant. Such a
controller is referred to as safe. This paper studies the regularity properties
of the minimum-norm safe controller as a stepping stone towards the design of
general continuous safe feedback controllers. We characterize the set of points
where the minimum-norm safe controller is discontinuous and show that it
depends solely on the safe set and not on the particular CBF that describes it.
Our analysis of the controller behavior as we approach a point of discontinuity
allows us to identify sufficient conditions to ensure it grows unbounded or it
remains bounded. Examples illustrate our results, providing insight into the
conditions that lead to (un)bounded discontinuous minimum-norm controllers.
