# Equivalence of Definitions of Differentiable Real Function at Point

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## Contents

## 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.