  Newtonian spaces generalize first-order Sobolev spaces to abstract metric
measure spaces. In this paper, we study regularity of Newtonian functions based
on quasi-Banach function lattices. Their (weak) quasi-continuity is
established, assuming density of continuous functions. The corresponding
Sobolev capacity is shown to be an outer capacity. Assuming sufficiently high
integrability of upper gradients, Newtonian functions are shown to be
(essentially) bounded and (H\"older) continuous. Particular focus is put on the
borderline case when the degree of integrability equals the "dimension of the
measure". If Lipschitz functions are dense in a Newtonian space on a proper
metric space, then locally Lipschitz functions are proven dense in the
corresponding Newtonian space on open subsets, where no hypotheses (besides
being open) are put on these sets.
