Algebrability and Riemann integrability of the composite function

Abstract

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.