Lineability of the functions that are Sierpiński–Zygmund, Darboux, but not connectivity

Abstract

Assuming that continuum $$\mathfrak {c}$$ is a regular cardinal, we show that the class $${{\,\mathrm{PES}\,}}{\setminus }{{\,\mathrm{Conn}\,}}$$ of all functions from $$\mathbb {R}$$ to $$\mathbb {R}$$ that are perfectly everywhere surjective (so Darboux) but not connectivity is $$\mathfrak {c}^+$$ -lineable, that is, that there exists a linear space of $$\mathbb {R}^\mathbb {R}$$ of cardinality $$\mathfrak {c}^+$$ that is contained in $$({{\,\mathrm{PES}\,}}{\setminus }{{\,\mathrm{Conn}\,}})\cup \{0\}$$ . Moreover, assuming additionally that $$\mathbb {R}$$ is not a union of less than $$\mathfrak {c}$$ -many meager sets, we prove $$\mathfrak {c}^+$$ -lineability of the class $${{\,\mathrm{SZ}\,}}\cap {{\,\mathrm{ES}\,}}{\setminus }{{\,\mathrm{Conn}\,}}$$ of Sierpinski-Zygmund everywhere surjective but not connectivity functions.