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When we learn calculus we usually:
1. Prove that polynomials, the exponential functions, the logarithmic functions, the trigonometric functions, the inverse trigonometric functions are continuous on its domain.
2. Prove that continuity of any two functions at a point will be preserved, in the resultant function, after taking finitely many times of the four arithmetic operations and composition.
∴ Elementary functions are continuous on its domain.

Question:
For every elementary function $f$, and every point $p$ in the domain of $f$, and every $\epsilon > 0$,
can we always find a $\delta > 0$ explicitly as a closed-form expression (including "the maximum function") in terms of $f$, $p$ and $\epsilon$ such that $| x - p | < δ \implies | f(x) - f(p) | < ε$ ?
(In other words, can we always construct at least one valid $\epsilon-\delta$ style argument showing $f$ satisfies the definition of being continuous at $p$?)

After all, elementary functions are infinitely many, but human beings only have finite amount and time.

At the moment the definition of "elementary functions" follows this webpage:
https://www.encyclopediaofmath.org/index.php/Elementary_functions

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    $\begingroup$ This is an interesting question, but I suspect it will be more appropriate at math.stackexchange. (I'm not voting to close, since it's possible there are some subtleties I'm not seeing at the moment.) $\endgroup$
    – Noah Schweber
    Commented Jun 29, 2018 at 21:13
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    $\begingroup$ Just to clarify: the downvote and closure vote (not mine) are re: this question's appropriateness for this site, not its inherent value. I think it would be much better received at MSE (although I think you should put some thought into what you mean by "closed form"). Note to others that the OP's earlier question on similar lines isn't quite the same as this one. $\endgroup$
    – Noah Schweber
    Commented Jun 29, 2018 at 21:35
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    $\begingroup$ It seems like this question is a bit contentious. It might be helpful if those in favor of it were to say a bit about why it's appropriate here ... $\endgroup$
    – Noah Schweber
    Commented Jun 29, 2018 at 22:57
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    $\begingroup$ Can you please clarify what you want when you ask for a “closed-form expression... in terms of $f$”? $\endgroup$
    – Zach Teitler
    Commented Jun 30, 2018 at 7:00
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    $\begingroup$ Isn't the answer 'obviously yes', because the proofs of continuity of products/compositions/linear-combinations each give closed-form bounds, and composing those closed-form bounds appropriately gives the closed-form bound for the elementary function in question? $\endgroup$
    – Adam P. Goucher
    Commented Jun 30, 2018 at 9:31

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