Families of feebly continuous functions and their properties

Abstract Let f : R 2 → R . The notions of feeble continuity and very feeble continuity of f at a point 〈 x , y 〉 ∈ R 2 were considered by I. Leader in 2009. We study properties of the sets F C ( f ) (respectively, V F C ( f ) ⊃ F C ( f ) ) of points at which f is feebly continuous (very feebly continuous). We prove that V F C ( f ) is densely nonmeager, and, if f has the Baire property (is measurable), then F C ( f ) is residual (has full outer Lebesgue measure). We describe several examples of functions f for which F C ( f ) ≠ V F C ( 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.