Boundary functions and sets of curvilinear convergence for continuous functions

Abstract

If ? is a function whose domain is a subset E of the set of curvilinear convergence off, then ? is called a boundary function for f if, and only if, for each x E E there exists an arc y at x such thatf(z) -+ ;(x) as z -* x along y. Let S be another metric space. We shall say that a function ? is of Baire class < 1(S, M) if (i) domain ? = S, (ii) range ? c M, and (iii) there exists a sequence {0n} of continuous functions, each mapping S into M, such that n, -* 0 pointwise on S. We shall say that ? is of honorary Baire class < 2(S, M) if (i) domain ? = S, (ii) range ? c M, and (iii) there exists a countable set Nc S and there exists a function b of Baire class _ 1(S, M) such that ?(x) = +(x) for every x E S-N. It is known that iff is a continuous function mapping D into the Riemann sphere, then the set of curvilinear convergence off is of type Fa,, and any boundary function forf is of honorary Baire class _ 2(C, Riemann sphere). (See [3], [4], [5], [6], [9].) J. E. McMillan [6] posed the following problem. If A is a given set in C of type F,6, and if ? is a function of honorary Baire class _2(A, Riemann sphere), does there always exist a continuous function f mapping D into the Riemann sphere such that A is the set of curvilinear convergence of f and ? is a boundary function for f? The purpose of this paper is to give an affirmative answer to McMillan's question. However, the corresponding question for real-valued functions remains open. (See Problems 1 and 2 at the end of this paper.) In proving our result, we first give a proof under the assumption that ? is a bounded complexvalued function, and we then use a certain device to transfer the theorem to the Riemann sphere. As we shall indicate in an appendix, the same device can be