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)