Lineability of Linearly Sensitive Functions

Abstract

AbstractIn the paper we will focus on lineability of some subsets of $${\mathbb {R}}^{\left[ 0,1\right] }$$R0,1 which are called linearly sensitive. A function f is called linearly sensitive with respect to the property (or condition) (P) if f has the property (P) and for any $$a\ne 0$$a≠0 the function $$f+a\cdot {{\,\mathrm{id}\,}}$$f+a·id does not have the property (P). We discuss some general method of proving $${\mathfrak {c}}$$c-lineability and use this method to examine lineability of the family of all continuous functions linearly sensitive to the Luzin (N)-property, the family of functions linearly sensitive to the Świątkowski condition and the family of functions linearly sensitive to the strong Świątkowski condition.