T.J. Kaczynski: Boundary functions for functions defined... Boundary functions for functions defined in a disk.
T.J. Kaczynski
J. Math. Mech. 14:589-612.
Let D denote the unit disk |z| < 1, C its boundary, and let f(z) be any function that is defined in D and takes its values in some metric space S. Then a boundary function for f is a function t on C such that for every x ( C there exists an arc v at x with
lim   f(z) = t(x).
z -> x
z ( v
The author proves several theorems on boundary functions in the following four cases: (1) f(z) a homeomorphism of D onto D, (2) f(z) a continuous function, (3) f(z) a Baire function and (4) f(z) a measurable function. These theorems include answers to two questions raised by Bagemihl and Piranian.

Theorem 1 states that if f(z) is a homeomorphism of D onto D, then there exists a countable set N such that t|C - N is continuous.

In the case of continuous functions, one needs some definitions. Let S and T be metric spaces. f is said to be of Baire class 1(S, T) if and only if (i) domain f = S, (ii) range f ( T and (iii) there exists a sequence {f(n)} of continuous functions, each mapping S into T, such that f(n) -> f pointwise on S. g is of honorary Baire class 2(S, T) if and only if (i) domain g = S, (ii) range g ( T and (iii) there exists a function f of Baire class 1(S, T) and a countable set N such that f|S - N = g|S - N. Using these defnitions, Theorems 2 and 3 read as follows. Theorem 2: Let f be a continuous real-valued function in D and let t be a finite-valued boundary function for f. Then t is of honorary Baire class 2(C, R), where R is the set of real numbers. Theorem 3: Let f be a continuous function mapping D into the Riemann sphere S and let t be a boundary function for f. Then t is of honorary Baire class 2(C, S).

In the cases of Baire functions and measurable functions, for the sake of convenience consider the open upper half-plane D0: I(z) > 0, and its boundary C0: I(z) = 0, instead of D and C, respectively. Theorem 4 states that if f is a real-valued function of Baire class a > 1 in D0, and t is a finite-valued boundary function, then t is of Baire class a + 1. As an immediate consequence of Theorem 4, one has Theorem 5: Let f be a real-valued Borel-measurable function in D0 and let t be a finite-valued boundary function for f; then t is Borel-measurable.

Next, the author proves that for an arbitrary function t on C0, there exists a function f on D0 such that f(z) = 0 almost everywhere and t is a boundary function for f. The paper concludes with some remarks concerning extensions of these theorems into three dimensions.

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