Definition:Differentiable Mapping/Real Function/Point

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Definition

Let $f$ be a real function defined on an open interval $\openint a b$.

Let $\xi$ be a point in $\openint a b$.


Definition 1

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

$\displaystyle \lim_{x \mathop \to \xi} \frac {\map f x - \map f \xi} {x - \xi}$

exists.


Definition 2

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

$\displaystyle \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$

exists.


These limits, if they exist, are called the derivative of $f$ at $\xi$.


Also see