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

Definition

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.

Sources

• 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 2.1$