D'Aniello, E.Fernández Sánchez, JuanMaiuriello, M.Seoane Sepúlveda, Juan Benigno2026-01-162026-01-16202410.1007/s13398-024-01677-xhttps://hdl.handle.net/20.500.14352/130482Prof. J. Fernández–Sánchez would like to dedicate this work to the memory of Antonio Carmona Reche (1962–2024). Prof. J.B. Seoane–Sepúlveda would like to dedicate this work to the memory of his aunt Carmen Sepúlveda Aramburu (1947–2024).In this note, we show that there exist a 2c-generated free algebra S ⊂ RR of Riemann integrable functions and a free algebra C ⊂ R[0,1] of continuous functions, having c-generators, such that r ◦ c is not Riemann integrable for any r ∈ S and c ∈ C. This result is the best possible one in terms of lineability within these families of functions and, at the same time, an improvement of a previous result ([6, Theorem 2.7]). In order to achieve our results we shall employ set theoretical tools such as the Fichtenholz-Kantorovich-Hausdorff theorem, Cantor-Smith-Volterra–type sets, and classical real analysis techniques.engjournal articlehttps://doi.org/10.1007/s13398-024-01677-xrestricted accessLineabilityAlgebrabilityBounded real functionMatemáticas (Matemáticas)Análisis funcional y teoría de operadores12 Matemáticas1202 Análisis y Análisis Funcional