function in $\mathbb{Q}$ with positive derivative A function $f: \mathbb{Q} \to \mathbb{Q}$ is differentiable over $\mathbb{Q}$ with $f'(q)$>$0$ for every $q$ in $\mathbb{Q}$.
Normaly when the derivative is positive for everything in the domain the function is rising. I'm looking for an counter example that this isn't always for a function defined in $\mathbb{Q}$.

Consider the function
$$f(q)=\begin{cases}q, & \textrm{if }q^3<2 \\ q-1, & \textrm{if }q^3\ge2.\end{cases}$$
We have
$$f'(q)=\lim_{h\to 0}\frac{f(q+h)-f(q)}{h}=1 \quad \forall q \in \mathbb{Q}.$$
However, $f(q)$ is not monotonically increasing.
