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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