#Contents# * table of contents {:toc} ## Definition ## Given a [[sequentially Cauchy complete Archimedean ordered field]] $\mathbb{R}$ and a [[real vector space]] $V$, a scalar function $f:V \to \mathbb{R}$ is a __directionally differentiable function__ if for every vector $v:V$ and line $L_v \subseteq V$, $$L_v \coloneqq \sum_{v:V} \Vert \sum_{a:\mathbb{R}} v = a w \Vert$$ with canonical equivalence $m:\mathbb{R} \cong L_v$ and canonical inclusion $i:L_v \hookrightarrow V$, the composite $f \circ i \circ m$ is a [[differentiable function]]. ## See also ## * [[differentiable function]] * [[directional derivative]]