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 eiT at which f has a finite or infinite limit along some open Jordan arc lying in the disk and having one endpoint at eiT. A boundary function of f is a function t defined on A such that each t(eiT) 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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