[[!redirects derivative]] #Contents# * table of contents {:toc} ## Definition ## Given an [[Archimedean ordered field]] $F$, a 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 $0 \lt \vert h \vert \lt \delta$ and for every element $x:F$, $$\vert f(x + h) - f(x) - h D(f)(x) \vert \lt \epsilon$$ ## See also ## * [[Archimedean ordered field]]