How does one determine with 100% certainty that a certain function can't ever possibly be put into terms of elementary functions, or really any arbitrary list of functions?
Take the gamma function for instance, . I am fairly confident the gamma function cannot be put into terms of elementary functions, but on the other hand, I've never ever seen anything remotely resembling a proof of that statement or any statement of that nature about any special function. How would I determine it can't be put into terms of other non-special functions or a mix of special functions and elementary functions?
I take it this question is related to your other one.
The sort of proof you are looking for is known as an existence and uniqueness theorem, of which there are several each pertaining to a particular area of mathematics.
It looks as though you are studying what is known as the functions of a real variable.
Even in this limited area the E&U theorems involve some highly abstract maths which fills large textbooks.
A few 'standards' are
Hurewicz Dimension Theory
Hobson The Theory of Functions of a Real Variable (2 vols)
Titchmarsh The Theory of Functions
Burkhill A second Course in Mathematical Analysis
Graves Theory of Functions of Real Variables
Verblunsky An introduction to the Theory of Functions of a Real Variable
Littlewood The Elements of the Theory of Real Functions
Some of these are rather old.