Given any differentiable function $f:\mathbb{R}\to \mathbb{R}$ and an open interval $(a,b)$ contained in the range of the derivative $g:=f'$ of $f$, does $g^{-1}((a,b))$ have to contain a non-empty interior. Of course, $g$ is not necessarily continuous. But it has some nice properties: For example, we know that the set of continuity points of $g$ is of second category in $\mathbb{R}$ (by an argument based on Baire category theorem). In addition, $g$ has the intermediate value property by Darboux's theorem. i.e., mapping any interval into an interval. BTW, it is not an exercise or statement from any book. So, I wouldn't be surprised that there is a counterexample for it.
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1So, using the properties you know, can you attempt to solve the problem? You should do it, not us. – GEdgar Jun 30 '17 at 19:53
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3@GEdgar I tried but can't find the answer or give a counter example. It is not an exercise. – Ice sea Jun 30 '17 at 19:56
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@DehanChen This is the key – Bettybel Jun 30 '17 at 19:58
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@DehanChen A more direct link to a proof (Wikipedia). – Bettybel Jun 30 '17 at 20:04
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@Bettybel But it is just Darboux's theorem. The question here is quite different from that. Or, you think one can follow the same argument? – Ice sea Jun 30 '17 at 20:07
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1So: try to show that any function with the Darboux property, regardless of whether it is a derivative, has your property. – GEdgar Jun 30 '17 at 20:19
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A Pompeiu derivative is a function that is the derivative of an everywhere differentiable function and that vanishes in a dense set. Taking $g$ to be a non-zero Pompeiu derivative taking some non-zero value $y$, the preimage of the open interval $(y-|y|,y+|y|)$ is non-empty but contains no interval.
Colin McQuillan
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This is of course wrong. Since $y$ should be such that $g$ takes that value but such that no other value inside $(y-|y|, y+|y|)$ is taken either. – Bettybel Jun 30 '17 at 20:44
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@Bettybel: I think you mean "every other value is also taken", which I didn't notice was a condition. It's true that particular interval might not be contained in the image of $g$, but by Darboux's theorem the image of a non-zero Pompeiu derivative contains some open interval (which can be taken not to include 0). – Colin McQuillan Jun 30 '17 at 20:57
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I just find an interesting consequence about the function you construct. If $y:=g(x_0)\neq 0$, then $g$ has to be discontinuous at $x_0$. Otherwise, the continuity of $g$ at $x_0$ implies that $g^{-1}((y-|y|,y+|y|))$ is a neighborhood of $x_0$. – Ice sea Jul 02 '17 at 12:08