# 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 $r : U-x \to \R$ such that:

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

#### 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 $r : U-x \to \R$ such that:

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

### 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 $\mathbb X$.