Definition:Differentiable Mapping/Real Function/Point

From ProofWiki
Jump to navigation Jump to search

Definition

Let $f$ be a real function defined on an open interval $\left({a \,.\,.\, b}\right)$.

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


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.


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


Also see