Introduction. A function f is a boundary function if f is defined on the real line and is the limit in some restricted sense of a function 4) defined on the open upper half-plane or if f is defined on the unit circle and is the limit in some restricted sense of a function 1 defined on the open unit disk. Bagemihl and Piranian [I] and Kaczynski [2] have studied boundary functions for the case where at each point of the boundary the limit of 1? is assumed to exist along some arc with one endpoint on the boundary. The author [5], [6] has studied boundary functions which are obtained by taking boundary limits relative to Stolz angles and approximate limits relative to Stolz angles. In this paper we assume that at each point of the boundary there are two arcs along which the limits of c) exist and are equal. We will let R denote the set of real numbers. DEFINITION. If xER, an arc at (x, 0) is a Jordan arc y with one endpoint at (x, 0) and with all other points lying in the open upper half-plane. DEFINITION. r is a bi-arc at (x, 0) if F is the union of two arcs at (x, 0) which are disjoint except for the point (x, 0). DEFINITION. Let 4b: R X (0, oo ) -R and f: R-*R. The function f is a bi-arc boundary function for 4 if for each xER there is a bi-arc rP, such that lim 4b(u, v) =f(x), as (u, v)->(x, 0), for (u, v)EJ'r. DEFINITION. Let f : R-R. The function f is of honorary Baire class two if there is a function g:R R-R such that g is of Baire class one and f (x) = g (x) except on a countable set.