Definition:Differentiable Mapping/Real Function/Point

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:

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

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.

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

Also see

• Equivalence of Definitions of Differentiable Real Function at Point
• Definition:Derivative of Real Function at Point