\begin{definition}[Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 1]
Let $U$ be an open subset of $\R^n$. 
Let $f: U \to \R$ be a real-valued function.
Let $x \in U$.
{{questionable|$\map f x + \alpha_1 h_1 + \cdots + \alpha_n h_n$ is a real number, so for the expression below to make sense, $\map r h\cdot h$ needs to be a real number, but $h$ is a vector and $\map r h$ is a real number, so $\map r h\cdot h$ is a vector (if $\cdot$ is supposed to denote scalar multiplication)}}
$f$ is '''differentiable at $x$''' {{iff}} there exist $\alpha_1, \ldots, \alpha_n \in \R$ and a real-valued function $r: U \setminus \set x \to \R$ such that:
:$(1):\quad \map f {x + h} = \map f x + \alpha_1 h_1 + \cdots + \alpha_n h_n + \map r h\cdot h$
:$(2):\quad \ds \lim_{h \mathop \to 0} \map r h = 0$
Category:Definitions/Differentiable Real-Valued Functions
\end{definition}