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Journal of Convex Analysis 30 (2023), No. 1, 065--080
Copyright Heldermann Verlag 2023

On Special Partitions of [0,1] and Lineability within Families of Bounded Variation Functions

Luis Bernal-González
Dep. de Análisis Matemático, Facultad de Matemáticas, Inst. de Matemáticas Antonio de Castro Brzezicki, Universidad de Sevilla, Spain

Juan Fernández-Sánchez
Universidad de Almería, Spain

Juan B. Seoane-Sepulveda
Inst. de Matemática Interdisciplinar, Dep. de Análisis Matemático y Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Spain

Wolfgang Trutschnig
Dept. of Mathematics, University of Salzburg, Austria

We show that there exist large algebraic structures (vector spaces, algebras, closed subspaces, etc.) formed entirely (except for 0), on one hand, by singular, nowhere monotonic functions on [0,1] and, on the other hand, by absolutely continuous nowhere monotonic functions. Several tools, of independent interest, related to obtaining special partitions of R into uncountable collections will be provided and used. The results obtained in this note are either new or improved versions of already existing ones.

Keywords: Lineability, spaceability, bounded variation function, singular function, absolutely continuous function, nowhere monotonic function.

MSC: 15A03, 46B87, 26A45, 26B30.

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