Editing Absolute value (section)

===Derivative===
The real absolute value function has a [[derivative]] for every {{math|''x'' ≠ 0}}, but is not [[differentiable]] at {{math|1=''x'' = 0}}. Its derivative for {{math|''x'' ≠ 0}} is given by the [[step function]]:<ref name="MathWorld">{{cite web| url = http://mathworld.wolfram.com/AbsoluteValue.html| title = Weisstein, Eric W. ''Absolute Value.'' From MathWorld – A Wolfram Web Resource.}}</ref><ref name="BS163">Bartle and Sherbert, p.&nbsp;163</ref>
:<math>\frac{d\left|x\right|}{dx} = \frac{x}{|x|} = \begin{cases} -1 & x<0 \\  1 & x>0. \end{cases}</math>

The real absolute value function is an example of a continuous function that achieves a [[Maximum and minimum|global minimum]] where the derivative does not exist.

The [[subderivative|subdifferential]] of&nbsp;{{math|{{abs|{{mvar|x}}}}}} at&nbsp;{{math|1=''x'' = 0}} is the interval&nbsp;{{closed-closed|−1, 1}}.<ref>Peter Wriggers, Panagiotis Panatiotopoulos, eds., ''New Developments in Contact Problems'', 1999, {{ISBN|3-211-83154-1}}, [https://books.google.com/books?id=tiBtC4GmuKcC&pg=PA31 p.&nbsp;31–32]</ref>

The [[complex number|complex]] absolute value function is continuous everywhere but [[complex differentiable]] ''nowhere'' because it violates the [[Cauchy–Riemann equations]].<ref name="MathWorld"/>

The second derivative of&nbsp;{{math|{{abs|{{mvar|x}}}}}} with respect to&nbsp;{{mvar|x}} is zero everywhere except zero, where it does not exist. As a [[generalised function]], the second derivative may be taken as two times the [[Dirac delta function]].