Lineability of linearly sensitive functions (Q2309626)

A function \(f:[0,1] \to \mathbb{R}\) is said to be linearly sensitive with respect to a property (P) if \(f\) has (P) but \(f + a\cdot \mathrm{id}\) does not have (P) for any \(a \in \mathbb{R}$, $a \neq 0\). The authors show the set of functions that enjoy linear sensitivity with respect to various properties is \(c\)-lineable. For instance, recall that a function \(f\) satisfies the Luzin (N)-property if \(f\) takes null sets to sets of measure \(0\). Let \(A_N\) be the collection of continuous \(f\) that are linearly sensitive with respect to (N). Theorem. \(A_N\) is \(c\)-lineable. Two examples are given involving the Świątkowski condition, which a function \(f\) satisfies provided that, for all \(x_1 < x_2\) such that \(f(x_1) \neq f(x_2)\), there is some point \(x \in (x_1,x_2)\) such that \(f\) is continuous at \(x\) and that \(f(x)\) is between \(f(x_1)\) and \(f(x_2)\). Theorem. The family \(A_{S}\) of all Baire one functions that are linearly sensitive with respect to the Świątkowski condition is \(c\)-lineable. The general method of proof uses the following result: Let \(M \subset \{G:[0,1] \to \mathbb{R}\}\). Suppose that there is a sequence of disjoint closed subsets \(([a_n,b_n]) \subset [0,1]\) and a sequence \((F_n:[a_n,b_n] \to \mathbb{R})\) such that, for any \(A \subset \mathbb{N}\) and any bounded sequence of non-zero real numbers \((\alpha_i)_{i \in A}\), the function \[G(x) = \alpha_iF_i(x) \text{ for }x \in [a_i,b_i],\ i \in A, \text{ and }G(x)=0\text{ otherwise}\] is in \(M\). Then \(M\) is \(c\)-lineable. Several interesting examples are provided.