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.