Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 2

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Definition

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

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

Let $x \in U$.


$f$ 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:

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