Editing Absolute value (section)

==Absolute value function==
[[Image:Absolute value.svg|thumb|360px|The [[graph of a function|graph]] of the absolute value function for real numbers]]
[[Image:Absolute value composition.svg|256px|thumb|[[composition of functions|Composition]] of absolute value with a [[cubic function]] in different orders]]
The real absolute value function is [[continuous function|continuous]] everywhere. It is [[differentiable]] everywhere except for {{math|1=''x'' = 0}}.  It is [[monotonic function|monotonically decreasing]] on the [[Interval (mathematics)|interval]] {{open-closed|−∞, 0}} and monotonically increasing on the interval {{closed-open|0, +∞}}. Since a real number and its [[additive inverse|opposite]] have the same absolute value, it is an [[even function]], and is hence not [[Inverse function|invertible]]. The real absolute value function is a [[piecewise linear function|piecewise linear]], [[convex function]].

For both real and complex numbers the absolute value function is [[idempotent]] (meaning that the absolute value of any absolute value is itself).

===Relationship to the sign function===
The absolute value function of a real number returns its value irrespective of its sign, whereas the [[sign function|sign (or signum) function]] returns a number's sign irrespective of its value. The following equations show the relationship between these two functions:
:<math>|x| = x \sgn(x),</math>
or
:<math> |x| \sgn(x) = x,</math>
and for {{math|''x'' ≠ 0}},
:<math>\sgn(x) = \frac{|x|}{x} = \frac{x}{|x|}.</math>

===Relationship to the max and min functions===
Let <math>s,t\in\R</math>, then the following relationship to the [[minimum]] and [[maximum]] functions hold:
:<math>|t-s|= -2 \min(s,t)+s+t</math>
and
:<math>|t-s|=2 \max(s,t)-s-t.</math>

The formulas can be derived by considering each case <math>s>t</math> and <math>t>s</math> separately.

From the last formula one can derive also <math>|t|= \max(t,-t)</math>.

===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]].

===Antiderivative===
The [[antiderivative]] (indefinite [[integral]]) of the real absolute value function is

:<math>\int \left|x\right| dx = \frac{x\left|x\right|}{2} + C,</math>

where {{mvar|C}} is an arbitrary [[constant of integration]]. This is not a [[complex antiderivative]] because complex antiderivatives can only exist for complex-differentiable ([[holomorphic]]) functions, which the complex absolute value function is not.

=== Derivatives of compositions ===
The following two formulae are special cases of the [[chain rule]]:

<math>{d \over dx} f(|x|)={x \over |x|} (f'(|x|))</math>

if the absolute value is inside a function, and

<math>{d \over dx} |f(x)|={f(x) \over |f(x)|} f'(x)</math>

if another function is inside the absolute value. In the first case, the derivative is always discontinuous at <math display="inline">x=0</math> in the first case and where <math display="inline">f(x)=0</math> in the second case.