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One relevant thing here is that you are referring to differentiating and integrating within the class of so-called elementary functions, which are built recursively from polynomials and complex expone …

Yeah, these force it to be ordinary differentiation. We have to show that for each fixed $x_0 \in \mathbb{R}$, the composite $$C^\infty(\mathbb{R}) \stackrel{L}{\to} C^\infty(\mathbb{R}) \stackrel{e …

I hesitate to answer again, but I agree with comments by Deane Yang and others that so far the discussions haven't quite gotten to the bottom of things. (Not that I promise to succeed in doing so now, …

I can't claim much knowledge here, but I am given to understand that the class of differentiable functions (or the class of functions which are derivatives of such) is really quite nasty and complicat …

In Andrew Gleason's interview for More Mathematical People, there is the following exchange concerning Gleason's work on Hilbert's fifth problem on whether every locally Euclidean topological group is …

After seeing wood's last comment (comment #2 under his question), I've decided to add a few words (a bit too many for a comment) which hopefully make clear the force of Qiaochu's answer. Generally s …

Note: This is a major rewrite of my earlier answer, to include necessary and sufficient conditions applicable to an even wider class of functions. Instead of expanding to the class of all analytic f …

This appears as Problem 4 of an MSRI Emissary publication, and indeed this problem has been discussed by Noam Elkies. See also his web page (problem 8 here), where he links to a solution. The conclu …

In one of my comments over at the other thread (this other thread), I had mentioned some discussion at the $n$-Category Café about a differential $\lambda$-calculus (from the looks of it, different to …

Comment by Cherng-tiao Perng converted to an answer: It appears that Theorem A of this paper solves your problem.

I will recount the more general statement of linear independence of characters, given in Lang's Algebra book, and credited to Artin. Let $G$ be a group, and $K$ a field. Then distinct homomorphisms $\ …

I don't know if this is of help, but I have seen this idea for defining integration elsewhere, specifically on pages 10-11 of Reed and Simon's Functional Analysis. It would go something like this: let …

It's probably not quite what you're looking for, but a necessary and sufficient condition for existence of elementary antiderivatives is given in this nice write-up on Dave Rusin's web, here. He inclu …