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Lineability, differentiable functions and special derivatives

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Abstract

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 map**s that do not satisfy Rolle’s theorem on real infinite dimensional Banach spaces. Several examples and graphics illustrate the obtained results.

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Notes

  1. Recall that a cone is a set endowed with two operations, addition and multiplication by positive scalars, which fulfill the usual properties (commutativity, associativity, existence of neutral element, etc.). If this set contains a linearly independent subset of infinite cardinality, the cone is said to be infinite. The dimension of the cone is the maximal possible cardinality of such a linearly independent set. In this paper, we shall say that a set M is strongly \(\mathfrak {c}\)-coneable whenever it contains (except for 0) a cone P of dimension \(\mathfrak {c}\) and this cone contains \(\mathfrak {c}\) algebraically independent elements as well (and, also, every non-trivial algebraic combination of elements of P, of positive coefficients, must belong to M).

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Acknowledgements

D.L. Rodríguez-Vidanes and J.B. Seoane–Sepúlveda were supported by Grant PGC2018-097286-B-I00. W. Trutschnig gratefully acknowledges the support of the WISS 2025 project ‘IDA-lab Salzburg’ (20204-WISS/225/197-2019 and 20102-F1901166-KZP)

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Authors and Affiliations

  1. Grupo de investigación “Teoría de cópulas y aplicaciones”, Universidad de Almería, Carretera de Sacramento s/n, 04120, Almería, Spain

    J. Fernández-Sánchez

  2. Departamento de Análisis y Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040, Madrid, Spain

    D. L. Rodríguez-Vidanes

  3. Departamento de Análisis y Matemática Aplicada, Facultad de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040, Madrid, Spain

    J. B. Seoane-Sepúlveda

  4. Department for Mathematics, University Salzburg, Hellbrunnerstrasse 34, 5020, Salzburg, Austria

    W. Trutschnig

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  1. J. Fernández-Sánchez
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  2. D. L. Rodríguez-Vidanes
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  3. J. B. Seoane-Sepúlveda
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Correspondence to J. B. Seoane-Sepúlveda.

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Communicated by M. S. Moslehian.

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Fernández-Sánchez, J., Rodríguez-Vidanes, D.L., Seoane-Sepúlveda, J.B. et al. Lineability, differentiable functions and special derivatives. Banach J. Math. Anal. 15, 18 (2021). https://doi.org/10.1007/s43037-020-00103-9

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