Impossibility of constructing continuous functions of $(n+1)$ variables from functions of $n$ variables by means of certain continuous operators

Continuous functions on a unit cube are considered. The concept of continuity regulator is introduced: in the definition of uniform continuity it governs the transition "from $\varepsilon$ to $\delta$". The problem of obtaining continuous functions of $(n+1)$ variables with continuity regulator $\delta$ variables with the same continuity regulator by means of uniformly continuous operators with continuity regulators that are superpositions of the regulator $\delta$ is posed. The insolubility of this problem is demonstrated for continuity regulators $\delta$ ($\varepsilon$) such that for each $\alpha\geqslant0$ the inequality $\delta(\varepsilon)\geqslant\varepsilon^{1+\alpha}$ holds starting from some $\varepsilon_\alpha$.