This shows that "elementary function" needs a good definition. We do NOT want to allow, for example $f(x) = 1$ when $x$ rational and $f(x) = -1$ when $x$ irrational. Even though $f^2 = 1$, this $f$ is not an algebraic function.
So, correctly defined, an elementary function is an analytic function on a domain in the complex plane, such that ...... [fill in the usual conditions]
Added later. My advice: For "elementary function" do not use the popularized form of the definition as in Wikipedia. Instead, use a definition from the actual mathematics papers. (Papers with proofs, not just quickie approximate definitions for the masses.)
For example
"Integration in Finite Terms", Maxwell Rosenlicht,
The American Mathematical Monthly 79 (1972), 963--972.
Stable URL: http://www.jstor.org/stable/2318066
Everything is carried out in differential fields ... In particular, every function involved is infinitely differentiable ... None of those "discontinuous elementary functions" mentioned in the question. Not even $|x| = \sqrt{x^2}$ is elementary.
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"Algebraic Properties of the Elementary Functions of Analysis", Robert H. Risch,
American Journal of Mathematics 101 (1979) 743--759.
Stable URL: http://www.jstor.org/stable/2373917
He also works in differential fields. Some quotes:
The elementary functions of a complex variable $z$ are those analytic functions that are built up from the rational functions of $z$ by successively applying algebraic operations, exponentiating, and taking logarithms. As is well known, this class includes the trigonometric and basic inverse trigonometric functions.
[Part II]
Suppose $\mathbb{C}(z, \theta_1, \dots, \theta_m) = \mathcal{D}_m$ is the abstract field, isomorphic to a field of meromorphic functions on some region $R$ of the complex plane, ...
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