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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.

Some additive Darboux-like functions

Ciesielski K.

Journal of Applied Analysis

1998

Vol. 4, nr 1

43-51

In this note we will construct several additive Darboux-like functions f: R −> R answering some problems from (an earlier version of) [4]. In particular, in Section 2 we will construct, under different additional set theoretical assumptions, additive almost continuous (in sense of Stallings) functions f: R −> R whose graph is either meager or null in the plane. In Section 3 we will construct an additive almost continuous function f: R −>R which has the Cantor intermediate value property but is discontinuous on any perfect set. In particular, such an f does not have the strong Cantor intermediate value property.