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 $h \in \mathbb{R}^n$.