This paper is devoted to the
analysis of sets in the plane which have at least one limit direction at each point. In
the proof of a variational technique used in univalent function theory Schiffer used
the fact that a continuum in the plane which only has limit directions ±1 is a
horizontal segment, a result of Haslam-Jones which used measure theory
and some rather unusual topological terminology. The attempt to find an
elementary topological proof has led to several false proofs in the literature. Not
only do we establish Haslam-Jones’ result using only elementary topology
but we obtain the conclusion under far weaker conditions allowing us to
obtain a set theoretic counterpart to the real variable theorem that says if the
upper right Dini derivative of a function is zero on an interval, then the
function is constant on that interval. We then extend the result to allow for
the possibility of exceptional points. Our strongest result gives a complete
classification of continua with exceptional points. The paper closes with an open
problem.
