Having a directional derivative

Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}^m$, let $a \in U$, let $v \in \mathbb{R}^n$, and let $L \in \mathbb{R}^m$. Let $g(t) = f(a + t v)$ for real $t$ near $0$. The function $f$ has the directional derivative $L$ at $a$ in the direction $v$ when $g$ is differentiable at $t = 0$ with $g'(0) = L$. Equivalently, $\dfrac{f(a + t v) - f(a)}{t} \to L$ as $t \to 0$.

Remarks

This definition states a property of $f$, $a$, $v$ and $L$. The directional derivative $D_v f(a)$ is then the vector $L$, which is unique. Only the values of $f$ on the line through $a$ in the direction $v$ matter. So $f$ can have a directional derivative in each direction at $a$ and still fail to be continuous at $a$, see this counterexample.

Explanations

L1 human

The function $g(t) = f(a + t v)$ is the function $f$ seen along the straight line through $a$ with direction $v$. To have a directional derivative means that this function of one variable has a derivative at the start of the line.