Definition:Differentiable Mapping/Real-Valued Function/Point

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

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

Let $x \in \mathbb X$.

Then $f$ is differentiable at $x$ if there exists a linear operator $T:\R^n \to \R$ such that
 * $\displaystyle f(x+h)=f(x) + Th + \epsilon(h)$

with $\epsilon(h)/|h| \to 0$ as $h \to 0$.

Also see

 * Characterization of Differentiability for clarification of this definition.