On superpositions of continuous functions defined on the Baire space (Q1582831)

The paper deals with the following question: if \(M(n)\) is a class of functions of \(n\) variables and \([M(n)]\) is the set of functions representable as superpositions of functions from \(M(n)\), is it true that \(M(n+1)\subset [M(n)]\)? Using the method based on Boolean functions introduced by the same author [Probl. Kibern. 37, 5-17 (1980; Zbl 0472.26006)], it is proved that for functions mapping \(I^n\) into \(I\), where \(I= \prod^\infty_{n=1} \{0,1\}\) is equipped with the metric of the Baire space, the above inclusion depends essentially on some kind of the modulus \(\varphi\) of continuity. The answer is positive in the case when \(\varphi(t)\geq (1+ a)t\) (Theorem 1) and negative if the last condition is violated (Theorem 2 and Theorem 3).