Definition:Left-Hand Derivative

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Definition

Let $B$ be a Banach space over the set of real numbers $\R$.

Let $f: \R \to B$ be a mapping from $\R$ to $B$.


The left-hand derivative of $f$ is defined as the left-hand limit:

$\displaystyle f'_- \left({x}\right) = \lim_{h \to 0^-} \frac {f \left({x + h}\right) - f \left({x}\right)} h$

If the left-hand derivative exists, then $f$ is said to be left-hand differentiable at $x$.


Real Functions

Let $f: \R \to B$ be a real function.


The left-hand derivative of $f$ is defined as the left-hand limit:

$\displaystyle f'_- \left({x}\right) = \lim_{h \mathop \to 0^-} \frac {f \left({x + h}\right) - f \left({x}\right)} h$

If the left-hand derivative exists, then $f$ is said to be left-hand differentiable at $x$.


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