That's basically it. I need to find a function on the interval $[0,1]$ that isn't a derivative of any derivable function.
I've found one possible solution which sets $0$ for every $x \in \mathbb{Q}$, and $1$ for every $x$ from $\mathbb{R}\setminus \mathbb{Q}$, but I don't really understand it and am not sure if it is correct.
yes, to expand on MathematicsStudent1122's reply, derivatives have the Darboux property (see Darboux's theorem on wiki), that is, if $f:I\to\mathbb{R}$ is differentiable in the interval $I$ and $f'$ takes two values, then it takes all the values in between. So the function $g$ which is $0$ on the rationals and $1$ on the irrationals cannot be the derivative of a function $f$ on any subinterval.
