Definition:Differentiable Mapping/Real Function/Interval

Definition

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 $\openint a b$.

On a Closed Interval

Let $f$ be differentiable on the open interval $\openint a b$.

If the following limit from the right exists:

$\ds \lim_{x \mathop \to a^+} \frac {\map f x - \map f a} {x - a}$

as well as this limit from the left:

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

then $f$ is differentiable on the closed interval $\closedint a b$.