I know we can't write every function as derivative o some other function. Darboux theorem gives easy way to find such example. And I just read absolutely continuous function which gives condition when a function can be written as derivative of some other function almost everywhere. But is there some necessary and sufficient condition for a function to be derivative of some other function.
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Related: http://math.stackexchange.com/questions/20578/why-is-integration-so-much-harder-than-differentiation – Arthur Dec 06 '14 at 14:26
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This should prove enlightening also. – David Mitra Dec 06 '14 at 14:37
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It's a hard problem and it's known as the derivatives characterization problem.
Take $f:\mathbb{R} \to \mathbb{R}$ to be a derivative, say $f=g'$.
Darboux's theorem establishes that $f$ must satisfy the intermediate value property. Because $f(x) = lim_{n\to \infty} n(g(x+\frac{1}{n}) - g(x))$ pointwise, then $f$ must be in the first class of Baire. This way you have two necessary conditions.
On the other hand, continuity is an example of a sufficient condition. If $f$ is continuous then by the fundamental theorem of calculus $f$ is the derivative of $g(x)=\int_{0}^{x}f(t)dt$. Of course $f$ needs not to be continuous to be a derivative. Nevertheless, the discontinuity set of $f$ can't be any possible set, for example, for being $f$ in the first class of Baire. There is a theorem characterizing the possible sets of discontinuity for derivatives: the Zahorski theorem
The problem is hard because you want to find a satisfactory set of necessary and sufficient conditions. See for example this book of Bruckner.
For a brief explanation of Baire classes you can read this.
