> 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.