Given an Archimedean ordered field$F$ , a functionpointwise continuous function$f:F \to F$ is$f:F \to F$ 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<\mathrm{max}|(h,|-h)<\delta $ 0 \lt \max(h, \vert -h) h \vert \lt \delta and for every element $x:F$,