Definition:Differentiable Mapping/Real Function

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Definition

At a Point

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 $\xi$.


On an Open Interval

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


Then $f$ is differentiable on $\left({a \,.\,.\, b}\right)$ if and only if $f$ is differentiable at each point of $\left({a \,.\,.\, b}\right)$.


On the Real Number Line

Let $f$ be a real function defined on $\R$.

By definition, $\R$ is an (unbounded) open interval.


Let $f$ be differentiable on the open interval $\R$.

That is, let $f$ be differentiable at every point of $\R$.


Then $f$ is differentiable everywhere (on $\R$).


Also see


Sources