Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 1

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

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

Let $x \in U$.

$f$ is differentiable at $x$ there exist $\alpha_1,\ldots,\alpha_n \in\R$ and a real-valued function $r : U \to \R$ such that:


 * $(1):\quad$ $\displaystyle f \left({x + h}\right) = f \left({x}\right) + \alpha_1h_1 + \cdots + \alpha_nh_n + r\left({h}\right)\cdot h$
 * $(2):\quad$ $\displaystyle\lim_{h\to0} r(h) = 0$.