The present work either extends or improves several results on lineability of differentiable functions and derivatives enjoying certain special properties. Among many other results, we show that there exist large algebraic structures inside the following sets of special functions: (1) The class of differentiable functions with discontinuous derivative on a set of positive measure, (2) the family of differentiable functions with a bounded, non-Riemann integrable derivative, (3) the family of functions from (0, 1) to $$\mathbb {R}$$ that are not derivatives, or (4) the family of mappings that do not satisfy Rolle’s theorem on real infinite dimensional Banach spaces. Several examples and graphics illustrate the obtained results.

中文翻译：

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线性、可微函数和特殊导数

目前的工作扩展或改进了具有某些特殊性质的可微函数和导数的线性化的几个结果。在许多其他结果中，我们表明以下特殊函数集内存在大型代数结构：（1）在一组正测度上具有不连续导数的可微函数类，（2）具有有界的可微函数族，非黎曼可积导数，(3) 从 (0, 1) 到 $$\mathbb {R}$$ 的函数族不是导数，或 (4) 不满足 Rolle 定理的映射族实无限维巴拿赫空间。几个例子和图形说明了所获得的结果。