  Verifying set invariance has classical solutions stemming from the seminal
work by Nagumo, and defining sets via a smooth barrier function constraint
inequality results in computable flow conditions for guaranteeing set
invariance. While a majority of these historic results on set invariance
consider flow conditions on the boundary, recent results on control barrier
functions extended these conditions to the entire set, although they required
regularity conditions on the barrier function. This paper fully characterizes
set invariance through \emph{minimal barrier functions} by directly appealing
to a comparison result to define a flow condition over the entire domain of the
system. A considerable benefit of this approach is the removal of regularity
assumptions of the barrier function. This paper also outlines necessary and
sufficient conditions for a valid differential inequality condition, giving the
minimum conditions for this type of approach. We also show when minimal barrier
functions are necessary and sufficient for set invariance.
