Definition:Differentiable Mapping/Real Function/Interval

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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:

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

as well as this limit from the left:

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

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


Sources