Equivalence of Definitions of Differentiable Real Function at Point

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Theorem

Let $I \subset \R$ be an open interval.

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


The following definitions of the concept of differentiable real function at a point are equivalent:

Definition 1

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.

Definition 2

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

$\displaystyle \lim_{h \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h$

exists.


Proof