Kolmogorov–Arnold representation theorem

Every multivariate continuous function \(f:[0,1]^n \to \mathbb{R}\) can be represented as a superposition of continuous single-variable functions. Namely

\[f(x) = f(x_1,\ldots, x_n) = \sum_{q=0}^{2n}\Phi_q\big(\sum_{p=1}^n \phi_{q,p}(x_p)\big)\]

where \(\phi_{q,p}:[0,1]\to \mathbb{R}\) and \(\Phi_q:\mathbb{R}\to\mathbb{R}\). We say that \(\phi_{q,p}\) are inner functions.

In particular there are proofs with specific constructions.

the non-smoothness of the inner functions and their "wild behavior" has limited the practical use of the representation