Definition:Differentiable Mapping/Real Function

Definition

Let $f$ be a real function defined on an open interval $\openint a b$.

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

$f$ 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.

$f$ 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 $\xi$.

Let $f$ be a real function defined on an open interval $\openint a b$.

Then $f$ is differentiable on $\openint a b$ if and only if $f$ is differentiable at each point of $\openint a b$.

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

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

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