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

中文翻译：

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Sierpiński-Zygmund、Darboux 函数的可线性性，但不是连通性

假设连续统 $$\mathfrak {c}$$ c 是常规基数，我们证明类 $${{\,\mathrm{PES}\,}}{\setminus }{{\,\mathrm{Conn }\,}}$$ PES \ 从 $$\mathbb {R}$$ R 到 $$\mathbb {R}$$ R 的所有函数的 Conn 都是完全满射的（所以 Darboux）但不是连通性是 $$ \mathfrak {c}^+$$ c + -lineable，即存在 $$\mathbb {R}^\mathbb {R}$$ RR 的基数 $$\mathfrak {c}^ 的线性空间+$$ c + 包含在 $$({{\,\mathrm{PES}\,}}{\setminus }{{\,\mathrm{Conn}\,}})\cup \{0\} $$ ( PES \ Conn ) ∪ { 0 } . 此外，另外假设 $$\mathbb {R}$$ R 不是小于 $$\mathfrak {c}$$ c -many meager set 的并集，我们证明 $$\mathfrak {c}^+$$ c + 类的线性度 $${{\,\mathrm{SZ}\,}}\cap {{\,\mathrm{ES}\,}}{\setminus }{{\,\mathrm{Conn} \,