Maximal lineability of the class of Darboux not connectivity maps on \(\mathbb{R}\) (Q2050198)

Let \(\kappa\) be a cardinal number. A subset \(M\) of a vector space \(X\) is said to be \(\kappa\)\emph{-lineable} (\emph{in} \(X\)) provided there exists a linear space \(Y\subset M\cup\{0\}\) of dimension \(\kappa\). Let \(\mathrm{Conn}\) be the class of all connectivity functions in \(\mathbb{R}^\mathbb{R}\), that is, having connected graphs (as subspaces of \(\mathbb{R}^2\)), while \(\mathrm{PES}\) is the family of all perfectly everywhere surjective maps \(f\colon\mathbb{R}\to\mathbb{R}\), that is, such that \(f[P]=\mathbb{R}\) for every non-empty perfect set \(P\subset\mathbb{R}\). The first result in the area of lineability was proved by \textit{V. I. Gurarij}. In 1966 [Sov. Math., Dokl. 7, 500--502 (1966; Zbl 0185.20203); translation from Dokl. Akad. Nauk SSSR 167, 971--973 (1966)] he showed that the set of continuous nowhere differentiable functions on \([0,1]\), together with the constant \(0\) function, contains an infinite-dimensional vector space, that is, it is \(\omega\)-lineable. Let \(c\) be the cardinality of \(\mathbb{R}\). In the present paper, the author shows that the family of all \(\mathrm{PES} \setminus \mathrm{Conn}\) mappings is \(2^c\)-lineable.