$\newcommand{\lax}{\operatorname{lax}}$ Liouville's theorem is well known and it asserts that:
The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions.
The problem I got from this is what is an elementary function? Who defines them? How do we define them?
Someone can, for example, say that there is a function which is called $\lax(\cdot)$ which is defined as:
$$ \lax\left(x\right)=\int_{0}^{x}\exp(-t^2)\mathrm{d}t. $$
Then, we can say that $\lax(\cdot)$ is a new elementary function much like $\exp(\cdot)$ and $\log(\cdot)$, $\cdots$.
I just do not get elementary functions and what the reasons are to define certain functions as elementary.
Maybe I should read some papers or books before posting this question. Should I? I just would like to get some help from you.