Families of feebly continuous functions and their properties

Abstract

Let f:R2→R. The notions of feeble continuity and very feeble continuity of f at a point 〈x,y〉∈R2 were considered by I. Leader in 2009. We study properties of the sets FC(f) (respectively, VFC(f)⊃FC(f)) of points at which f is feebly continuous (very feebly continuous). We prove that VFC(f) is densely nonmeager, and, if f has the Baire property (is measurable), then FC(f) is residual (has full outer Lebesgue measure). We describe several examples of functions f for which FC(f)≠VFC(f). Then we consider the notion of two-feeble continuity which is strictly weaker than very feeble continuity. We prove that the set of points where (an arbitrary) f is two-feebly continuous forms a residual set of full outer measure. Finally, we study the existence of large algebraic structures inside or outside various sets of feebly continuous functions.