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On symmetrically porouscontinuous functions

Kowalczyk S., Turowska M.

Journal of Applied Analysis

2016

Vol. 22, nr 2

141--151

In the present paper, we introduce the notion of symmetrically porouscontinuous functions. We investigate some properties of symmetric porouscontinuity and its connections with the notion of porouscontinuity, studied by Borsík and Holos in [2]. We prove that there are 2c symmetrically porouscontinuous functions, which extends results of [1] concerning ρ-upper continuous functions.

Exceptional sets for universally polygonally approximable functions

Evans M. J., Humke P. D.

Journal of Applied Analysis

2001

Vol. 7, nr 2

175-190

In [9], the present authors and Richard O'Malley showed that in order for a function be universally polygonally approximate it is necessary that for each ε > 0, the set of points of non-quasicontinuity be σ - (1 - ε ) symmetrically porous. The question as to whether that condition is sufficient or not was left open. Here we prove that if a set, E = U∞n=1 En, such that each Ei is closed and 1-symmetrically porous, then there is a universally polygonally approximable function, f, whose set of points of non-quasicontinuity is precisely E. Although it is tempting to call this a partial converse to our earlier theorem it might be more since it is not known if these two notions of symmetric porosity differ in the class of F? sets.

Contrasting symmetric porosity and porosity

Evans M. J., Humke P. D.

Journal of Applied Analysis

1998

Vol. 4, nr 1

19-41

It is known that the following two fundamental properties of porosity fail for symmetric porosity: 1) Every nowhere dense set A contains a residual subset of points x at which A has porosity 1. 2) If A is a porous set and 0 < p < 1, then A can be written as a countable union of sets, each of which has porosity at least p at each of its points. Here we explore the somewhat surprising extent to which these properties fail to carry over to the symmetric setting and investigate what symmetric analogs do hold.