Two fixed functions can approximate any continuous function using only addition and composition

Abstract

We prove that two fixed univariate functions, namely, an arbitrary continuous non-affine function and a particular affine function, are sufficient to approximate continuous functions of one variable under the operations of addition and composition. The same fixed functions can also be used to approximate multivariate continuous functions, provided that the coordinate functions are also available. We also show that the number of generators can be reduced from two to one. We construct a specific continuous function that generates a dense class in the univariate setting and, together with the coordinate functions, in the multivariate setting.