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

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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 left at $b$, but not both.


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