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

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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.


\(\ds \map {f'_+} 0\) \(=\) \(\ds 1\) 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.


Demonstrated in Derivative of Absolute Value Function.