In the paper we will focus on lineability of some subsets of $${\mathbb {R}}^{\left[ 0,1\right] }$$ R 0 , 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.

中文翻译：

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线性敏感函数的线性度

在本文中，我们将重点讨论 $${\mathbb {R}}^{\left[ 0,1\right] }$$ R 0 , 1 的一些子集的线性敏感性，它们被称为线性敏感。如果 f 具有属性 ( P ) 并且对于任何 $$a\ne 0$$ a ≠ 0 函数 $$f+a\cdot ，则函数 f 被称为对属性（或条件）（ P ）线性敏感{{\,\mathrm{id}\,}}$$ f + a · id 没有属性 ( P )。我们讨论了证明 $${\mathfrak {c}}$$ c -lineability 的一些一般方法，并使用这种方法来检查所有对 Luzin ( N )-property 线性敏感的连续函数族的线性度，函数族对 Świątkowski 条件线性敏感，而函数族对强 Świątkowski 条件线性敏感。