Elementary and integral-elementary functions

By an integral-elementary function we mean any real function that can be ob tained from the constants, sin x, e(x), log x, and arcsin x (defined on (-1 , 1)) using the basic algebraic operations, composition and integration. Th e rank of an integral-elementary function f is the depth of the formula def ining f. The integral-elementary Functions of rank less than or equal to n are real-analytic and satisfy a common algebraic differential equation P-n( f, f',..., f((k))) = 0 with integer coefficients. We prove that every continuous function f: R --> R can be approximated unif ormly by integral-elementary functions of bounded rank. Consequently, there exists an algebraic differential equation with integer coefficients such t hat its everywhere analytic solutions approximate every continuous function uniformly. This solves a problem posed by L. A. Rubel. Using the same basic functions as above, but allowing only the basic algebr aic operations and compositions, we obtain the class of elementary function s. We show that every differentiable function with a derivative not exceedi ng an iterated exponential can be uniformly approximated by elementary func tions of bounded rank. If we include the function arcsin x defined on [-1, 1], then the resulting class of naive-elementary functions will approximate every continuous function uniformly. We also show that every sequence can be uniformly approximated by elementar y functions, and that every integer sequence can be represented in the form f(n), where f is naive-elementary.