  In the last years the attention towards topological dynamical properties of
highly discontinuous maps has increased significantly. In [D.Corona, A. Della
Corte. The critical exponent functions. Comptes Rendus Math\'ematique, 360(G4),
315-332, 2022], a class of densely discontinuous interval maps, called
"critical exponent maps", was introduced. These maps are defined through the
word-combinatorics concept of critical exponent applied to the binary expansion
of reals and show highly chaotically properties as well as some challenging
problems. In this paper we identify an error in the proof of Theorem 7 in
[D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus
Math\'ematique, 360(G4), 315-332, 2022], a purely combinatorial result which in
fact does not hold. We show that most of the results in [D.Corona, A. Della
Corte. The critical exponent functions. Comptes Rendus Math\'ematique, 360(G4),
315-332, 2022], obtained there through Theorem 7, can be recovered. Moreover,
we propose as a conjecture a weaker form of Theorem 7.
