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If $f(x)$ is an elementary function and $f(x) = g(x) + h(x)$

Does that necessarily mean that both $g(x)$ and $h(x)$ are elementary functions ?

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    $\begingroup$ obviously not if $g(x)=u(x)-h(x)$ with $u(x)$ elementary and $h(x)$ not elementary. $\endgroup$ Mar 9 '18 at 14:52
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No, if $f(x)$ is the sum of an elementary function $h(x)$ with a non elementary function $g(x)$, then $f(x)+(-g(x))$ is the sum of two non elementary functions that are elementary.

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No. Suppose that $g(x)$ is a non-elementary function. Now consider $h(x)=-g(x)$ then $h$ is non-elementary as well, but $f=0$ and so $f$ is obviously elementary.

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