T.J. Kaczynski: Boundary functions and sets of curvilinear...
**Boundary functions and sets of curvilinear convergence for continuous functions.**

T.J. Kaczynski

*Trans. Amer. Math. Soc.* 141:107-125.

The author completes the investigation, initiated by Bagemihl and Piranian,
of boundary functions of continuous complex-valued functions defined in the
open unit disk D. the set of curvilinear convergence A of such a function
f is defined to be the set of those e^{iT} at which f has a finite or infinite
limit along some open Jordan arc lying in the disk and having one endpoint
at e^{iT}. A boundary function of f is a function t defined on A such that
each t(e^{iT}) is one of these limit values. The author proved that t differs
from some function of the first Baire class at at most countably many points,
and McMillan proved that A is of type F(sd). By means of an intricate
construction, the author proves that for any set A on the unit circle of
type F(sd), and for any function t defined on A that differs from some
function of the first Baire class at at most countably many points, there
exists a continuous complex-valued function f defined in D having A as its
set of curvilinear convergence and having t as its boundary function. The
author points out the the problem remains open for real-valued functions.

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