Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 2

Definition

Let $U$ be an open subset of $\R^n$.

Let $f: U \to \R$ be a real-valued function.

Let $\norm \cdot $ denote the Euclidean norm on $\R^n$.

Let $x \in U$.

$f$ is differentiable at $x$ if and only if there exists a linear transformation $T: \R^n \to \R$ and a real-valued function $r: U \setminus \set x \to \R$ such that:

$(1): \quad \map f {x + h} = \map f x + \map T h + \map r h \norm h$

$(2): \quad \ds \lim_{h \mathop \to 0} \map r h = 0$