# Homotopy Type Theory differentiable function > history (Rev #8, changes)

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# Contents

## 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 $x:F$,

$\vert f(x + h) - f(x) - h D(f)(x) \vert \lt \epsilon$