  The Abstract Boundary singularity theorem was first proven by Ashley and
Scott. It links the existence of incomplete causal geodesics in strongly
causal, maximally extended spacetimes to the existence of Abstract Boundary
essential singularities, i.e., non-removable singular boundary points. We give
two generalizations of this theorem: the first to continuous causal curves and
the distinguishing condition, the second to locally Lipschitz curves in
manifolds such that no inextendible locally Lipschitz curve is totally
imprisoned. To do this we extend generalized affine parameters from $C^1$
curves to locally Lipschitz curves.
