I guess you're right, I was probably mislead this whole time. I went through my old analysis class book [1] and there doesn't seem to be an explicit definition of elementary functions. The best I can find is this paragraph (I translate from italian):

> The elementary functions of analysis, that is powers, roots, exponentials, logarithms and their inverses, functions obtained from the former by arithmetic operations or composition, admit the limit f(p) for x → p, for any p in their set of definition. The study of such functions, which is not limited to the sole real functions of real variable, is carried out naturally in the setting of metric spaces.

That said, I'm relatively sure that a definition was given in class and it didn't include arbitrary roots: despite being notoriously difficult, the exam didn't require students to draw the graph of any elementary function including implicitly-defined algebraic roots.

I picked up another one of the old recommended books [2] and it seems to be similarly vague; while the book currently taught in my university [3], gives this definition:

> The following functions (from ℂ to ℂ) are called the elementary functions of the Analysis:

> 1) Rational functions (integral or fractional)

> 2) Algebraic functions (explicit or implicit)

> 3) The exponential function

> 4) The logarithm function

> 5) All those functions that can be obtained by combining a finite number of times the functions of kind 1)...4).

So, roots of arbitrary polynomials implicitly defined are indeed considered elementary. I never knew this.

[1]: https://search.worldcat.org/title/1261811544

[2]: https://search.worldcat.org/title/801297519

[3]: https://search.worldcat.org/title/935666878

So, I did a bit of research and I wasn't going crazy: there are apparently two competing definitions of "elementary" in use [1]:

> the class of functions [...] is what I would call exponential-logarithmic functions or EL functions; that is, they are the functions that can be expressed using some finite combination of constant functions, the identity function, exp, log, composition, and arithmetic operations (+−×÷). Some authors call this class of functions elementary functions, but that term is now more commonly used in a different sense, which includes algebraic functions.

Evidently my professor was in the exponential-logarithmic camp.

[1]: https://mathoverflow.net/a/442656