Real functions Functions of several real variables Theorem

Kolmogorov–Arnold representation theorem

Statement

Let $n \ge 2$ . There are continuous functions $\varphi_{q,p} \colon [0, 1] \to \R$ , for $0 \le q \le 2n$ and $1 \le p \le n$ , with the following property. For every continuous function $f \colon [0, 1]^n \to \R$ there are continuous functions $\Phi_q \colon \R \to \R$ , for $0 \le q \le 2n$ , such that $f(x_1, \dots, x_n) = \sum_{q=0}^{2n} \Phi_q\left( \sum_{p=1}^{n} \varphi_{q,p}(x_p) \right) \quad \text{for all } (x_1, \dots, x_n) \in [0, 1]^n.$

Remarks

The inner functions $\varphi_{q,p}$ do not depend on $f$ ; only the outer functions $\Phi_q$ do. The inner functions can be chosen increasing. So the only genuine function of several variables needed to build every continuous function of $n$ variables is addition. Example: for $f(x, y) = xy$ on $[0, 1]^2$ there is an explicit formula of the same kind, $xy = \frac{1}{4}\left((x + y)^2 - (x - y)^2\right)$ , with fewer terms than the theorem uses. In Hilbert's thirteenth problem, Hilbert conjectured that some continuous functions of three variables are not superpositions of continuous functions of two variables. The theorem refutes this conjecture: every continuous function of several variables is a superposition of continuous functions of one variable and addition.

Going deeper

For a student

The theorem belongs to the study of superpositions of functions. It contrasts with results of Vitushkin for smooth functions: superpositions of $C^k$ functions of fewer variables do not give every $C^k$ function of $n$ variables. The inner functions are continuous but quite irregular, so the representation is not a practical way to compute $f$ , though it has inspired work on approximation by networks.

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Continuous function concept
Kolmogorov–Arnold representation theorem statement

Tag	`VQXDB9T838`
URI	`m2://VQXDB9T838`
Kind	statement, theorem
Writing	W0 stub
MSC	26B40
Status	This copy of the site has no build data. Run `m2 status VQXDB9T838`.
About	Continuous function
Uses	Continuous function
Same as	Wikidata Q25099402 Wikipedia Kolmogorov–Arnold representation theorem
Changed	2 November 2026

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