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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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superAnnoyingUser
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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 09 '13 at 20:33
  • Well let me add the "soft qustion" tag then. – superAnnoyingUser Dec 09 '13 at 20:34
  • That doesn't answer the question! What do you mean by "we don't know"? – Nick Peterson Dec 09 '13 at 20:35
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    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 09 '13 at 20:38
  • My question is well posed and rigorous. I'm looking for a function we can prove to be differentiable, but whose derivative we do not know. Does that answer your question, Mr. @Peterson? – superAnnoyingUser Dec 09 '13 at 20:39
  • No. If you know that it is differentiable, then you know its derivative: it is $f'(x):=\lim_{h\rightarrow0}(f(x+h)-f(x))/h$. Maybe you don't know how to express it in terms of elementary functions; is that what you're getting at? – Nick Peterson Dec 09 '13 at 20:40
  • I see the trivial answer now, thank you for pointing that out. I will edit accordingly. – superAnnoyingUser Dec 09 '13 at 20:43
  • @NicholasR.Peterson If $f$ is known at all points and that limit exists, is it obvious that it can be computed? – user7530 Dec 09 '13 at 20:51
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    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 http://math.stackexchange.com/questions/112067/how-discontinuous-can-a-derivative-be. 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 09 '13 at 21:02
  • I don't understand the question; but an antiderivative of a non elementary function does have a non elementary derivative, doesn't it? – egreg Dec 09 '13 at 21:29
  • Well, if $f(x)$ is elementary i.e. it is composed of algebraic, trigonometric, logarithmic and exponential functions then $f'(x)$ is also elementary because of the rules of derivatives. Hence any example of the kind you are seeking has to be a non-elementary function like say $\Gamma(x)$ or any of elliptic functions like $\text{sn}(x, k)$ for $0 < k < 1$. – Paramanand Singh Dec 10 '13 at 09:13

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

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