I have no idea about the seemingly obvious problem below.
Problem. If a continuous function $f(x)$ is differentiable on the set of real numbers, and its derivative is always zero when $x\in \mathbb{Q}$. Does that mean $f(x)$ is a constant function?
According to the Darboux theorem, we all know that the derivative function is not necessarily continuous, but it has the intermediate value property. I guess the answer is yes, but I really don't have any idea to solve the problem.
Can anyone help me?
