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  "body": "\n## Statement\n\nLet $n \\ge 2$.\nThere are [[continuous functions|0X3WGX95FE]] $\\varphi_{q,p} \\colon [0, 1] \\to \\R$, for $0 \\le q \\le 2n$ and $1 \\le p \\le n$, with the following property.\nFor 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\n$$f(x_1, \\dots, x_n) = \\sum_{q=0}^{2n} \\Phi_q\\left( \\sum_{p=1}^{n} \\varphi_{q,p}(x_p) \\right) \\quad \\text{for all } (x_1, \\dots, x_n) \\in [0, 1]^n.$$\n\n## Remarks\n\nThe inner functions $\\varphi_{q,p}$ do not depend on $f$; only the outer functions $\\Phi_q$ do.\nThe inner functions can be chosen increasing.\nSo the only genuine function of several variables needed to build every continuous function of $n$ variables is addition.\nExample: 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.\nIn Hilbert's thirteenth problem, Hilbert conjectured that some continuous functions of three variables are not superpositions of continuous functions of two variables.\nThe theorem refutes this conjecture: every continuous function of several variables is a superposition of continuous functions of one variable and addition.\n\n## Explanations\n\n### L1 human\n\nAny continuous function of many variables can be built from functions of one variable and the operation of adding numbers.\nFirst apply fixed one-variable functions to each coordinate and add the results; then apply one more one-variable function and add again.\nThe fixed inner functions work for every target function at once.\n\n### L2 human\n\nThe theorem belongs to the study of superpositions of functions.\nIt 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.\nThe 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.\n",
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