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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 $L_v \subseteq V$,

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