Let D be the open unit disk and let K be the unit circle. We say that α is an arc at ζ ∈ K if α is contained in D and is the image of a continuous function z = z(t) (0 ≤ t < 1) such that z(t) → ζ as t → 1. We call α a segment as ζ if the function z = z(t) is linear in t. If P is a property which is meaningful for each point of K, we say that nearly every point of K has property P if the exceptional set is a set of first Baire category in K. We assume that the reader is familiar with the rudiments of cluster set theory, and in particular with the terms ambiguous point, Meier point, and Plessner point of a function (cf. [4] or [7]).