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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... –  Nicholas R. 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"? –  Nicholas R. Peterson Dec 9 '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 9 '13 at 20:38
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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 math.stackexchange.com/questions/112067/…. 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

1 Answer 1

up vote 2 down vote accepted

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

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