Definition:Differentiable Mapping/Real-Valued Function/Point

Definition
Let $\mathbb X$ be an open subset 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 transformation $T:\R^n \to \R$ such that:


 * $\displaystyle f \left({x + h}\right) = f \left({x}\right) + T h + \epsilon \left({h}\right)$

with $\dfrac {\epsilon \left({h}\right) } {\left\vert{h}\right\vert} \to 0$ as $h \to 0$.

Also see

 * Characterization of Differentiability for clarification of this definition.