# 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:

- $\displaystyle \map {f'_-} x = \lim_{h \mathop \to 0^-} \frac {\map f {x + h} - \map f x} h$

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

### Real Functions

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

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

- $\displaystyle \map {f'_-} x = \lim_{h \mathop \to 0^-} \frac {\map f {x + h} - \map f x} h$

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

## Also known as

Some sources give this as the **left derivative**.