Given an Archimedean ordered field$F$, a pointwise continuous function$f:F \to F$ is pointwise differentiable if it comes with a function $D(f):F \to F$ called the derivative and such that for every positive element $\epsilon:F_+$, there exists a positive element $\delta:F_+$ such that for every element $h:F$ such that $0 \lt \max(h, -h) \lt \delta$ and for every element $x:F$,