> If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does.

I just looked through many of the best known real analysis texts, and not a single one defines them this way. This list included the texts by

Royden, Terence Tao, Rudin, Spivak, Bartle & Sherbert, Pugh, and a few others....

Can you cite a single text book that has this definition you claim is in every real analysis course? I find all evidence points to the opposite.

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