Abstract
We show that if Φ is an arbitrary countable set of continuous functions of n variables, then there exists a continuous, and even infinitely smooth, function ψ(x1,...,xn) such that ψ(x 1, ...,x n ) ≢g [ϕ (f 1(x 1, ... ,f f (x n ))] for any function ϕ from Φ and arbitrary continuous functions g and fi, depending on a single variable.
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Translated from Matematicheskie Zametki, Vol. 16, No. 4, pp. 517–522, October, 1974.
In conclusion, the author thanks G. P. Gavrilov for his statement of the problem and for his constant help.
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Agrachev, A.A. On superpositions of continuous functions. Mathematical Notes of the Academy of Sciences of the USSR 16, 897–900 (1974). https://doi.org/10.1007/BF01104251
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DOI: https://doi.org/10.1007/BF01104251