Vol. 74, No. 1, 1978

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Continua in the plane with limit directions

Douglas Michael Campbell and Jack Wayne Lamoreaux

Vol. 74 (1978), No. 1, 37–46

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.

Mathematical Subject Classification 2000
Primary: 54F15
Received: 7 May 1977
Published: 1 January 1978
Douglas Michael Campbell
Jack Wayne Lamoreaux