Definition:Differentiable Mapping/Real-Valued Function

At a Point
Let $f: \mathbb X \to \R$ be a real-valued function, where $\mathbb X \subseteq \R^n$.

Let $x = ( x_1, \ldots, x_n) \in \mathbb X$.

We say that $f$ is differentiable at $x$ with respect to $x_i$ if
 * $\displaystyle \lim_{h \to 0}{\frac{f(x_1,\ldots,x_i+h,\ldots,x_n) - f(x_1,\ldots,x_i,\ldots,x_n)}{h}}$

exists and is finite.

That is, if the partial derivative $\displaystyle\frac{\partial f}{\partial x_i}$ exists.

We say that $f$ is differentiable at $x$ if $f$ is differentiable at $x$ with respect to $x_i$ for each $i=1,\ldots,n$.

In a Region
Let $S \subseteq \mathbb X$.

Then $f$ is differentiable in a region $S$ iff $f$ is differentiable at each $x$ in $S$, and write $f \in \mathcal C^1 \left({S, \R}\right)$.

Also see

 * Characterization of Differentiability for clarification of this definition.