Abstract
1. Kolmogorov proved [1] that the set of functions of two variables representable as a certain combination of continuous functions of one variable and addition is everywhere dense in the space C(E2) of continuous functions defined on the square E2. It follows immediately from our result proved below that this is not true for the simplest combinations: the set of functions of the form χ[φ(x) + ψ(y)] even turns out to be nowhere dense in C(E2).
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References
Kolmogorov, A.N.: On the representation of continuous functions of several variables as a superposition of continuous functions of a smaller number of variables. Dokl. Akad. Nauk SSSR 108, No, 2 (1956).
Krondov, A.S.: On functions of two variables. Usp. Mat. Nauk 5, No, 1 (1950).
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(2009). On the representation of functions of two variables in the form χ[φ(x) + ψ(y)]. In: Givental, A., et al. Collected Works. Vladimir I. Arnold - Collected Works, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01742-1_1
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DOI: https://doi.org/10.1007/978-3-642-01742-1_1
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