Definition:Differentiable Mapping/Real Function/Point/Definition 1

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Definition

Let $f$ be a real function defined on an open interval $\left({a \,.\,.\, b}\right)$.

Let $\xi$ be a point in $\left({a \,.\,.\, b}\right)$.


Then $f$ is differentiable at the point $\xi$ if and only if the limit:

$\displaystyle \lim_{x \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$

exists.


Sources