In this note, we show that there exist a \(2^\mathfrak {c}\) -generated free algebra \(\mathcal {S} \subset \mathbb {R}^\mathbb {R}\) of Riemann integrable functions and a free algebra \(\mathcal {C} \subset \mathbb {R}^{[0,1]}\) of continuous functions, having \(\mathfrak {c}\) -generators, such that \(r \circ c\) is not Riemann integrable for any \(r \in \mathcal {S}\) and \(c \in \mathcal {C}\) . 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.
