I have the following question:
I have a function $f: \mathbb R \to \mathbb R$ which is differentiable everywhere.
I also have a set $G\subset\mathbb R$ which is dense in $\mathbb R$ and a $G\_\delta$-set.
I know that $f'(x)=0 \forall x\in G$. 


Can I conclude that $f$ is constant?


The answer is yes if $f'$ is continuous, but unfortunately I don't know that.


Many thanks!



Derivative is Zero on a dense G_delta set
Since the zero set is always a $G\_\delta$, the question is whether it being dense implies that the function is constant. This is false, and a counterexample is known as a [Pompeiu derivative](https://en.m.wikipedia.org/wiki/Pompeiu_derivative).


