Definition:Differentiable Mapping/Real-Valued Function/Point

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

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

Let $x \in U$.

Then $f$ is differentiable at $x$ there exists a linear transformation $T:\R^n \to \R$ and a real-valued function $r : U \to \R$ such that:


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

Also see

 * Characterization of Differentiability for clarification of this definition.