# One-Sided Derivative/Examples/Absolute Value Function at Zero

## Examples of One-Sided Derivatives

Let $f$ be the real function defined as:

$\map f x = \size x$

where $\size x$ denotes the absolute value function.

Then:

 $\ds \map {f'_+} 0$ $=$ $\ds 1$ where $\map {f'_+} 0$ denotes the right-hand derivative of $f$ at $x = 0$ $\ds \map {f'_-} 0$ $=$ $\ds -1$ where $\map {f'_-} 0$ denotes the left-hand derivative of $f$ at $x = 0$

while the derivative of $f$ at $x = 0$ does not exist.

## Proof

Demonstrated in Derivative of Absolute Value Function.

$\blacksquare$