Essential Cluster Sets Research Articles

Cluster sets for sobolev functions and quasiminimizers

In this paper, we study cluster sets and essential cluster sets for Sobolev functions and quasiharmonic functions (i.e., continuous quasiminimizers). We develop their basic theory with a particular ...

INTERSECTION PROPERTIES OF DIRECTIONAL ESSENTIAL CLUSTER SETS

Jarnik in 1936 proved a remarkable property of directional cluster sets. This result states that for a function \(f\) defined on the open upper half plane to the extended real line, each pair of directional cluster sets intersect at all points on the real line but a countable set of points. An example was constructed to show that the exact analogue of Jarnik’s result fails for directional essential cluster sets. Here we shall establish a certain variant of this analogue for directional essential cluster sets of measurable functions.

On Sums of Directional Essential Cluster Sets

On Sums of Directional Essential Cluster Sets

Derectional Cluster Sets and Directional Essential Cluster Sets

Derectional Cluster Sets and Directional Essential Cluster Sets

Intersection of essential cluster sets

Intersection of essential cluster sets

The Equality of Unilateral Derivates

C. J. Neugebauer has shown that if f is a continuous function of bounded variation defined on the real line, then the set E where the upper right derivate differs from the upper left derivate is of measure zero and first category. Here it is shown that this result is best possible; that is, given any measure zero first category set K, there is a continuous function of bounded variation for which $K \subseteq E$. It is also shown that if f is monotone, then E is $\sigma$-porous. This result can be used to provide an elementary proof of the fact that for an arbitrary function f the left and right essential cluster sets are identical except at a $\sigma$-porous set of points, a result first proved by L. Zajíček.

The equality of unilateral derivates

C. J. Neugebauer has shown that if f is a continuous function of bounded variation defined on the real line, then the set E where the upper right derivate differs from the upper left derivate is of measure zero and first category. Here it is shown that this result is best possible; that is, given any measure zero first category set K, there is a continuous function of bounded variation for which K ⊆ E K \subseteq E . It is also shown that if f is monotone, then E is σ \sigma -porous. This result can be used to provide an elementary proof of the fact that for an arbitrary function f the left and right essential cluster sets are identical except at a σ \sigma -porous set of points, a result first proved by L. Zajíček.

Intersection of sectorial cluster sets and directional essential cluster sets

Intersection of sectorial cluster sets and directional essential cluster sets

On Directional Essential Cluster Sets

On Directional Essential Cluster Sets

Cluster sets and essential cluster sets of meromorphic functions.

Cluster sets and essential cluster sets of meromorphic functions.

Normal Functions Have Connected Essential Cluster Sets

Normal Functions Have Connected Essential Cluster Sets

On essential cluster sets

On essential cluster sets

Strong essential cluster sets.

Strong essential cluster sets.

Essential cluster sets

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.