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3 Journal of Applied Analysis

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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

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.

Universally polygonally approximable functions

Evans. M. J., Humke P. D., O`Malley R. J.

Journal of Applied Analysis

2000

Vol. 6, nr 1

25-45

It has recently been established that any Baire class one function f : [0,1] -> R can be represented as the pointwise limit of a sequence of polygonal functions whose vertices lie on the graph of f. Here we investigate the subclass of Baire class one functions having the additional property that for every dense subset D of [0,1], the first coordinates of the vertices of the polygonal functions can be chosen from D.

Covering Baire 1 functions with darboux functions and the cofinality of the ideal of meager sets

Jordan F.

Journal of Applied Analysis

1999

Vol. 5, nr 1

71-94

Let Dar stand for the Darboux Baire class 1 functions. We show that the cofinality of the meager sets in R is the smallest cardinality of a set of Baire class 1 functions F such that for any finite collection of Baire class 1 functions G there is an f ∈ F such that f + G ⊆ Dar. Other results of this type are shown. These results are then considered as statements about additivity. The notion of super-additivity is introduced.