T.J. Kaczynski: On a boundary property... On a boundary property of continuous functions.
T.J. Kaczynski
Michigan Math. J. 13:313-320.
The author generalizes the result of McMillan (1966) to the effect that the set of curvilinear convergence of a continuous function f from D into Z is of type F(sd). The generalization considers f as a continuous function from D into a compact metric space E. Topologizing the set of closed sets C(E) of E with the Hausdorff metric and letting E be any closed set in C(E), it is shown that the set of all x ( C such that there is a boundary path v at x with the cluster set of f along v contained in some set of E is of type F(sd). Taking E to be the set of all singletons {y}, y ( E (which is closed in C(Z)) McMillan's theorem is obtained.

Various other corollaries are given by selecting appropriate closed sets E ( C(E).


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