Porosity in Spaces of Darboux-Like Functions

Abstract

It is known that the six Darboux-like function spaces of continuous, extendable, almost continuous, connectivity, Darboux, and peripherally continuous functions f : R → R, with the metric of uniform convergence, form a strictly increasing chain of subspaces. We denote these spaces by C, Ext, AC, Conn, D, and PC, respectively. We show that C and D are porous and AC and Conn are not porous in their successive spaces of this chain.1. f ∈ PC if the graph of f is bilaterally dense in itself.2. f ∈ D if f (J) is connected for each connected set J ⊂ R.

Porous and Boundary Sets in Darboux-Like Function Spaces

“…The following chart, in which → indicates proper inclusion, was lifted from [10] and [6]. In [13], we determine which spaces are porous or boundary sets for the chain of spaces of functions f : R → R in the top row of the chart. We indicate in the rest of the chart which function spaces are determined in this paper to be porous or boundary sets.…”