By using the mean value theorem (for which we require completeness) we can show for a function $f\colon [a,b]\to\mathbb{R}$ differentiable on $(a,b)$ that $$f \;\text{monotone increasing}\iff \forall x\in (a,b)\colon f'(x)\ge 0.$$ Now, differentiation does not require completeness. Limits can be defined by only using the rationals, thus we can define the derivative for a function only defined on $[a,b]\cap \mathbb{Q}$.
Then we can easily find counterexamples to the mean value theorem by choosing a differentiable function $f\colon[a,b]\to\mathbb{R}$ with $f(a)=f(b)$ whose only maximum lies in an irrational point and then considering the rational function $f\mid_{[a,b]\,\cap\, \mathbb{Q}}$.
My question is now: Is there a counterexample to the above statement, where the implication from the RHS to the LHS is incorrect? A function on the rationals or some "rational interval", which is not monotone, but differentiable everywhere with positive derivative?
If $f(q)=\frac q2$ when $q^2 < 2$ and $f(q)=q-2$ for $q^2 \ge 2$ with $q \in \mathbb Q$
then $f'(q) = \frac12$ or $1$ for all $q \in \mathbb Q$ but $f(q)$ is not an increasing function: $f(1)=\frac12$ while $f(2)=0$