Essential cluster sets

Abstract

1. In 1928, W.H. Young [6] showed that an arbitrary real function of a real variable has a remarkable symmetry property. Let f: R -> R be arbitrary. A number y is a right limit of f at x E R if there is a sequence xn > x, n = 1,... such that lim xn = x and lim f(x,,) = y. A left limit is defined similarly. The theorem of Young asserts that for every f the set of right limits is the same as the set of left limits for every x E R A, where A is a countable set. Young obtained an analogous result for functions of two variables. Here the role of right and left limits is assumed by limits within sectors with the point in question as vertex. Each sector yields a limit set. The result is that all the sectorial limit sets are the same at every p E R2A, where A is of planar measure zero and first category, but not necessarily countable. In 1938, H. Blumberg [2] obtained a variety of results along the lines of those obtained by Young. He pointed out, for example, that in the planar case, if the angles of the sectors considered exceed 1800, then the countable character of the exceptional set A is restored. Returning to the original theorem of Young, Zahorski asked if the result still holds if sets of density zero are neglected. L. Belowska [1] has shown that this is false by constructing a function where in this case the exceptional set is uncountable. However, M. Kulbacka [4] has shown that the exceptional set is always of the first category and measure zero. A simple proof of these results was given by C. Goffman [3]. It is natural to ask whether there are theorems similar to the ones of Young and Blumberg in higher dimensions, but with sets of density zero neglected. This paper gives an affirmative answer to this question by showing that a theorem analogous to the two dimensional result of Young holds if one disregards sets of density zero. In this connection, various examples are given relating to the structure of the exceptional set.