T.J. Kaczynski: Boundary functions for bounded...
Boundary functions for bounded harmonic functions.
Trans. Amer. Math. Soc. 137:203-209.
A function p(e) defined on the unit circle is a boundary function for a
function f(z) defined in the unit disk provided for each e, f(z) has the
limit p(e) at e along some curve lying in the unit disk and having one
endpoint at e. Any two boundary functions for the same function f
differ at only countably many points by the ambiguous-point theorem
of Bagemihl; and a boundary function for a continuous function differs
from some function in the first Baire class at only countably many points.
In answer to a question of Bagemihl and Piranian, the author constructs
a bounded harmonic function having a boundary function that is not in the
first Baire class. He shows that nevertheless the set of points of
discontinuity of such a boundary function is a set of the first Baire
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