# Definition:Left-Hand Derivative

## 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.