#Contents# * table of contents {:toc} ## Definition ## Given a calculus field $F$ and a $F$-[[calculus vector space]] $V$, a scalar function $f:V \to F$ is a __directionally differentiable function__ if for every vector $v:V$ and line $L_v \subseteq V$, $$L_v \coloneqq \sum_{v:V} \Vert \sum_{a:F} v = a w \Vert$$ with canonical equivalence $m:F \cong L_v$ 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]]