By an integral-elementary function we mean any real function that can be obtained from the constants, sin $x$, $e^{x}$, $\log x$, and arcsin $x$ (defined on ($-1,1$)) using the basic algebraic operations, composition and integration. The rank of an integral-elementary function $f$ is the depth of the formula defining $f$. The integral-elementary functions of rank $\leq n$ are real-analytic and satisfy a common algebraic differential equation $P_{n} (f,f',\ldots,f^(k))=0$ with integer coefficients.
¶ We prove that every continuous function $f:\mathbf{R} \rightarrow \mathbf{R}$ can be approximated uniformly by integral-elementary functions of bounded rank. Consequently, there exists an algebraic differential equation with integer coefficients such that 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 algebraic operations and compositions, we obtain the class of elementary functions. We show that every differentiable function with a derivative not exceeding an iterated exponential can be uniformly approximated by elementary functions 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 elementary functions, and that every integer sequence can be represented .in the form $f(n)$, where $f$ is naive-elementary.