Can there exist a non-constant continuous function that has a derivative of zero everywhere? Somebody told me that there exists a continuous function with a derivative of zero everywhere that is not constant.
I cannot imagine how that is possible and I am starting to doubt whether it's actually true. If it is true, could you show me an example? If it is not true, how would you go about disproving it?

Since there are no restrictions on the domain, it is actually possible. Let $f:(0,1)\cup(2,3)\to \mathbb R$ be defined by $f(x)=\left\{
 \begin{array}{ll}
  0  & \mbox{if } x \in (0,1) \\
  1 & \mbox{if } x\in (2,3)
 \end{array}
\right.$
