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One of the basis results in real analysis is the Darboux's theorem, which says the derivative of a differentiable function has the intermediate value property.

I've always been a bit dissatistied with this result as I think the following stronger result holds:

Let $f:[a,b]\to\mathbb R$ be a differentiable function, then the graph of $f'$ is connected.

Is there a proof of this statement?

(differentiable on $[a,b]$ means differentiable on $(a,b)$ and $f'_+(a)$, $f'_-(b)$ exist)

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I believe I have some references that you can use to cobble together a proof.

According to wiki, the derivative of any differentiable function is of Baire class 1.

Theorem 4.1 in this paper by Bruckner, Ceder (1965) (page 99) proves that every Darboux function of Baire class 1 has a connected graph.

As you have stated, we already know that the derivative is a Darboux function.

I'm not overly familliar with these concepts myself, so I can't vouch for the validity, say, the wiki claim, but this certainly should be a line of inquiry worth investigating.

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  • $\begingroup$ Nice find! It seems like a possible way. I'll leave this open for now in case someone wants to explain this in detail or finds a direct way. I'm also not too familiar with these concepts. $\endgroup$ – user2345215 Mar 6 '14 at 15:53
  • $\begingroup$ Sure. I might try to read through some of this material and produce a synthesis later, if I find time, as these concepts are semi-related to a course that I'm currently taking. $\endgroup$ – Joshua Pepper Mar 6 '14 at 15:55

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