A.A. Agrachev, On superpositions of continuous functions

Title: On superpositions of continuous functions
Authors: A.A. Agrachev
Journal title: Mathematical Notes
Year: 1974
Issue: 4
Volume: 16
Pages: 897–900
Citation:

A. A. Agrachev, “On superpositions of continuous functions”, Mat. Zametki, 16:4 (1974), 517–522

Abstract:

We show that if $$\Phi$$ is an arbitrary countable set of continuous functions of $$n$$ variables, then there exists a continuous, and even infinitely smooth, function $$\psi(x_1, …, x_n)$$ such that $$\psi(x_1, …, x_n) \neq g [\phi (f_1(x_1), …, f_n(x_n)]$$ for any function $$\phi$$ from $$\Phi$$ and arbitrary continuous functions $$g$$ and $$f_i$$ depending on a single variable.

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