\begin{definition}[Definition:Differentiable Mapping/Real Function/Interval/Closed Interval]
Let $f$ be a real function defined on a closed interval $\closedint a b$.
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 $\closedint a b$'''.
Similar definitions for differentiability on a half-open interval can be expressed for a real function which has either a limit from the right at $a$ or a limit from the left at $b$, but not both.
\end{definition}