# Kolmogorov–Arnold representation theorem

Kolmogorov–Arnold representation theorem In real analysis and approximation theory, the Arnold–Kolmogorov representation theorem (or superposition theorem) states that every multivariate continuous function can be represented as a superposition of the two-argument addition and continuous functions of one variable. It solved a more constrained, yet more general form of Hilbert's thirteenth problem.[1][2][3] The works of Vladimir Arnold and Andrey Kolmogorov established that if f is a multivariate continuous function, then f can be written as a finite composition of continuous functions of a single variable and the binary operation of addition.[4] More specifically, {displaystyle f(mathbf {x} )=f(x_{1},ldots ,x_{n})=sum _{q=0}^{2n}Phi _{q}left(sum _{p=1}^{n}phi _{q,p}(x_{p})right)} .