Can all functions be expressed in terms of elementary functions? After being introduced to the non-elementary function through an attempt to evaluate $\int x \tan (x)$, an interesting question occurred to me:
Can the non-elementary functions be decomposed to elementary ones?  For instance, the logarithm, an elementary, can be decomposed into multiplication (e.g. $\ln x=y$ is the same as $y$ iterations of $e*e$), another elementary.  So, is this decomposition possible to transform a complicated non-elementary function into an elementary one that can be easily evaluated?
 A: No. Doesn't matter which logical language you try to use and the interpretation, number of function that can be defined using that language is at most the number of finite string that can be formed using the symbols of that language. Since symbols set is finite, number of possible string is countable. Number of function however is far from countable, it is actually $c^{c}$ for function $\mathbb{R}\rightarrow\mathbb{R}$.
(in other word, even if you have way more than just elementary function, there are function you simply cannot describe at all)
EDIT: thanks for the comments. In light of these, I will add a few more variant:
-If we allow the symbols set to be of cardinality $c$ (say, maybe we allow not just elementary function, but any continuous function), and restrict to only measurable function. Then this is still impossible by the counting argument. Unfortunately, restriction to measurable function does not decrease cardinality, and increasing the symbols set cardinality only give you the cardinality of possible sentence to be $c$.
-If we allow the symbol set to be of cardinality $\aleph_{0}$, and restrict to continuous function, then it is still impossible by counting argument. Continuous function have cardinality $c$, but number of possible string is still $\aleph_{0}$.
A: It seems you mean a very general kind of "decomposition" in which you are allowed to rewrite the functions in terms of some procedure involving some sequence of "elementary" operations.
It seems to me that any "decomposition" in any reasonable sense would have to be capable of being described using a finite number of words and symbols. This would allow even procedures with an infinite number of steps converging on the answer, such as power series, as long as the rules for the procedure are finite.
It also seems reasonable that we can only have a fixed finite set of elementary operations to work with. Otherwise it's hard to see how they can all be elementary.
Given the possibility of writing any finite set of rules using a fixed finite set of operations, words, and symbols, there are only a countably infinite number of possible "decompositions" that can be written. (I'm assuming there is no limit on the length of a "decomposition"; if there were such a limit, the number of "decompositions" would be finite.)
But there are an uncountable number of functions from $\mathbb{N}$ to $\mathbb{N}$, let alone functions from $\mathbb{R}$ to $\mathbb{R}$.  Therefore we cannot decompose all functions.
