Definition:Differentiable Mapping/Real Function/Interval/Closed Interval

Definition
Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$. Let $f$ be differentiable on the open interval $\left({a \,.\,.\, b}\right)$.

If the following limit from the right exists:


 * $\displaystyle \lim_{x \to a^+} \frac {f \left({x}\right) - f \left({a}\right)} {x - a}$

as well as this limit from the left:
 * $\displaystyle \lim_{x \to b^-} \frac {f \left({x}\right) - f \left({b}\right)} {x - b}$

then $f$ is differentiable on the closed interval $\left[{a \,.\,.\, b}\right]$.

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 right at $b$, but not both.