Having a gradient

Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, let $a \in U$, and let $g \in \mathbb{R}^n$. The function $f$ has the gradient $g$ at $a$ when $$\lim_{h \to 0} \frac{f(a + h) - f(a) - g \cdot h}{\|h\|} = 0,$$ where the limit is over $h \ne 0$ with $a + h \in U$. Equivalently, $f$ is differentiable at $a$ and $Df(a)(h) = g \cdot h$ for each $h \in \mathbb{R}^n$.

Remarks

This definition states a property of $f$, $a$ and $g$. When it holds, the vector $g$ is unique, and it is the gradient $\nabla f(a)$. The same definition works in each real inner product space, with the inner product in place of the dot product.

Explanations

L1 human

The function has the gradient $g$ at $a$ when the plane through $(a, f(a))$ with slopes given by $g$ hugs the graph of $f$ near $a$. The error of this approximation must be small compared with the distance from $a$.