Every real continuous function on a compact square can be represented as , with continuous one-variable functions and fixed inner functions independent of the represented function. Grid-separated additive coordinates give the inner functions. For a residual of norm , assign at a sample point divided by three to each separated rectangle image, and continuously interpolate between images. Each point lies in at least three rectangles, so if summands are accurate, the error is at most . Choose oscillation to obtain contraction . Iteration gives outer functions by uniform convergence of corrections bounded by . Whole-plane reduction for continuous superposition removes compactness of the original domain without assuming boundedness of the original function.
For , choose a positive continuous function with for . Such an is obtained by piecewise linear interpolation of a sufficiently large increasing sequence. Then tends uniformly to zero at infinity. Transport to by and extend it by zero on the boundary; the extension is continuous. Apply the compact Kolmogorov-Arnold representation theorem, then multiply by . Multiplication introduces no additional primitive operation because uses only addition and continuous one-variable squaring and scaling.
For five pairs of continuous functions on , one can arrange that at every arbitrarily fine scale there are closed interval families whose product rectangles cover each point at least three times, and within each family the images under are mutually disjoint. To see this, put gaps near five shifted fine grids; gap sets for different shifts are disjoint. Each coordinate misses at most one family, so a pair misses at most two. Approximate any prescribed pair of functions by constants on the complementary intervals, interpolating across gaps. Perturb the finitely many constants to avoid all equalities between distinct pair sums. The rectangle images are then separated by a positive distance, a property preserved under sufficiently small uniform perturbations. For each required mesh size this property defines a dense open subset of . The Baire category theorem gives a tuple satisfying every scale simultaneously.
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The Kolmogorov–Arnold representation theorem, also known as the Kolmogorov–Arnold function representation theorem, is a result in the theory of multivariate functions that provides a way to express any continuous multivariate function as a superposition of continuous functions of fewer variables.