Algebrability and Riemann integrability of the composite function

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 precious result ([Theorem 2.7]{A}). 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.