Wyniki wyszukiwania - BazTech

1

Collections of Darboux-like, Baire one functions of two variables

Evans M. J., Humke P. D.

Journal of Applied Analysis

|

2010

|

Vol. 16, nr 1

135-149

EN

The collection of Baire class one, Darboux functions from [0,1] to R is a rich class of functions that has been intensely investigated and characterized over the years. Which Baire class one, real-valued functions defined on [0,1] x [0,1] form the most "natural" extension of this class to the two-variable setting is debatable, with many suggestions having been advanced. In light of recent interest in Darboux-like properties for derivatives (i.e. gradients) of differentiable functions of two variables, it seems that now is a good time to consider some of the most feasible notions of "Darboux-like" and investigate the relationships between them.

2

Almost everywhere first-return recovery

Evans M. J., Humke P. D.

Bulletin of the Polish Academy of Sciences. Mathematics

|

2004

|

Vol. 52, nr 2

185-195

EN

We present a new characterization of Lebesgue measurable functions; namely, a function f : [0,1] --> R is measurable if and only if it is first-return recoverable almost everywhere. This result is established by demonstrating a connection between almost everywhere first-return recovery and a first-return process for yielding the integral of a measurable function.

3

Exceptional sets for universally polygonally approximable functions

Evans M. J., Humke P. D.

Journal of Applied Analysis

|

2001

|

Vol. 7, nr 2

175-190

EN

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.

4

Contrasting symmetric porosity and porosity

Evans M. J., Humke P. D.

Journal of Applied Analysis

|

1998

|

Vol. 4, nr 1

19-41

EN

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.