-Sets of numbers are often presented in terms if types of equations that require such numbers in order to be solved.
-For example, we need integers to solve (for $x$) $ a+x = c$ when $c$ is less than $a$. And we need rationals to solve (for $x$) $ax = b$ , when $\frac ba$ is not an integer. Etc.
-Hence the idea of doing the same thing for functions. I mean, the idea of presentng, so to say, the genealogy of functions in terms of types of equations they allow to solve ( with an increasing degree of complexity).
-For example, we need " radical functions" ( and sometimes the absolute value function) to solve equations of the form : $x^n = a $.
We need logarithm functions to solve exponential equations.
These are just examples.
I recently heard about a mysterious "Lambert" function allowing to solve equations such as $ 2^x=x$.
- My aim, in asking this question, is to know whether we could gain a systematic classification of functions by proceeding in this fashion.
Such a system (sort of) exists. A function is polynomial if it is a polynomial in (one or several) variables; it is rational if it is given by the quotient of polynomials; it is algebraic if it can be expressed as the root of a polynomial (i.e., for example $y x^2 + y^3 + x = 0$ defines $y$ as a function of $x$, and viceversa); then there are trascendental functions that aren't algebraic. Of those, the functions $e^x$, $\ln x$, $\sin x$, $\cos x$, their rational combinations and their inverses are considered elementary trascendentals (you know them from calculus), higher trascendental functions are typically defined as integrals that aren't elementary (like $\operatorname{li}(x) = \int_0^x \frac{d t}{\ln t}$ or $\Gamma(z) = \int_0^\infty x^{z - 1} e^x \, d x$ when $z$ is not an integer) or by non-elementary solutions to differential equations (look up e.g. the Bessel functions).
But all of those are (mostly) "well-behaved" functions (continuous, differentiable at most points). They are a tiny part of all the functions, which include strange beasts like Weierstrass' function (continuous everwhere, nowhere differentiable), Thomae's function (continuous at all irrational $x$, discontinuous at rational $x$), Conway's base 13 function (everywhere discontinuous). Even a seemingly innocent equation like Cauchy's functional equation $f(x) + f(y) = f(x + y)$ has bizarre solutions (no, the obvious $f(x) = c x$ for some $c$ isn't the only type of solution).
This leads to very deep depths... way over my head, to be sure. An understandable (to a relative layperson, i.e., college calculus covered) introduction to some of the subtleties you'll find in Dunham's "Calculus Gallery" (Princeton University Press, 2008).