# Definition:Differentiable Mapping/Real-Valued Function

## Definition

### At a Point

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

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

Let $x \in U$.

#### Definition 1

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

$(1):\quad \map f {x + h} = \map f x + \alpha_1 h_1 + \cdots + \alpha_n h_n + \map r h\cdot h$
$(2):\quad \displaystyle \lim_{h \mathop \to 0} \map r h = 0$

#### Definition 2

$f$ 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:

$(1):\quad \map f {x + h} = \map f x + \map T h + \map r h \cdot h$
$(2):\quad \displaystyle \lim_{h \mathop \to 0} \map r h = 0$

### In an Open Set

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

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

Then $f$ is differentiable in the open set $\mathbb X$ if and only if $f$ is differentiable at each point of $\mathbb X$.

## Also see

• Results about differentiable real-valued functions can be found here.