  We discuss two types of `regularity point', points of continuity and R-points
for Banach function algebras, which were introduced by the first author and
Somerset in an earlier paper on non-regularity for Banach function algebras. We
show that, even for the natural uniform algebras $R(X)$ (for compact plane sets
X), these two types of regularity point can be different. We then give a new
method for constructing Swiss cheese sets $X$ such that $R(X)$ is not regular,
but such that $R(X)$ has no non-trivial Jensen measures. The original
construction appears in the first author's previous work. Our new approach to
constructing such sets is more general, and allows us to obtain additional
properties. In particular, we use our construction to give an example of such a
Swiss cheese set $X$ with the property that the set of points of discontinuity
for $R(X)$ has positive area.
