Definition:Differentiable Mapping/Real Function/Interval

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Definition

Let $f$ be a real function defined on an open interval $\left({a \,.\,.\, b}\right)$.


Then $f$ is differentiable on $\left({a \,.\,.\, b}\right)$ if and only if $f$ is differentiable at each point of $\left({a \,.\,.\, b}\right)$.


On a Closed Interval

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 $\left[{a \,.\,.\, b}\right]$.


Sources