Equivalence of Definitions of Differentiable Real Function at Point

Theorem

Let $\openint a b \subset \R$ be an open interval.

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

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

Definition 1

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

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

exists.

Definition 2

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

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

exists.

Proof

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