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Do we know of any differentiable function whose derivative is not an elementary function? This may be a silly question, but in the light of this answer, as pointed in the comments, finding an example may be pedagogical.

More importantly, can we prove the existence or non-existence of such a function?

Edit: An answer that is not in the form $f(x)=\int_0^x g(x)$ would be much appreciated. The point is to find an example that would be of value for the above answer.

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What do you mean by "we don't know"? That seems more like a philosophical question than anything else... – Nick Peterson Dec 9 '13 at 20:33
Well let me add the "soft qustion" tag then. – Student Dec 9 '13 at 20:34
That doesn't answer the question! What do you mean by "we don't know"? – Nick Peterson Dec 9 '13 at 20:35
How about this: for a differentiable function $f$ and point $p$, call $P_{q,\epsilon}$ the decision problem of whether or not $\|f'(p)-q\| < \epsilon.$ Is $P_{q,\epsilon}$ decidable for all $q,\epsilon$? – user7530 Dec 9 '13 at 20:38
I added a comment to your other post as well. Dave Renfro goes over some general theory in his answer here that is related to this…. The typical derivative in the sense of Baire's theorem is discontinuous on a co-measure zero $F_\sigma$ set, but still satisfies the intermediate value property. It is difficult to imagine writing down a formula for such a function (I know of no explicit examples), but in the Baire sense they are typical. – Chris Janjigian Dec 9 '13 at 21:02
up vote 2 down vote accepted

$$f(x)=\int _0 ^x \text{Erf}.$$

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