Editing Sign function (section)

=== Differentiation ===

The signum function <math>\sgn x</math> is [[Differentiable function|differentiable]] everywhere except when <math>x=0.</math> Its [[derivative]] is zero when <math>x</math> is non-zero:
<math display="block"> \frac{\text{d}\, (\sgn x)}{\text{d}x} = 0 \qquad \text{for } x \ne 0\,.</math>

This follows from the differentiability of any [[constant function]], for which the derivative is always zero on its domain of definition. The signum <math>\sgn x</math> acts as a constant function when it is restricted to the negative [[Interval (mathematics)#Definitions and terminology|open region]] <math>x<0,</math> where it equals {{math|−1}}. It can similarly be regarded as a constant function within the positive open region <math>x>0,</math> where the corresponding constant is {{math|+1}}. Although these are two different constant functions, their derivative is equal to zero in each case.

It is not possible to define a classical derivative at <math>x=0</math>, because there is a discontinuity there.

Although it is not differentiable at <math>x=0</math> in the ordinary sense, under the generalized notion of differentiation in [[distribution (mathematics)|distribution theory]], 
the derivative of the signum function is two times the [[Dirac delta function]]. This can be demonstrated using the identity <ref>{{MathWorld |title=Sign |id=Sign}}</ref>
<math display="block"> \sgn x = 2 H(x) - 1 \,,</math>
where <math>H(x)</math> is the [[Heaviside step function]] using the standard <math>H(0)=\frac{1}{2}</math> formalism.
Using this identity, it is easy to derive the distributional derivative:<ref>{{MathWorld |title=Heaviside Step Function |id=HeavisideStepFunction}}</ref>
<math display="block"> \frac{\text{d}\sgn x}{\text{d}x} = 2 \frac{\text{d} H(x)}{\text{d}x} = 2\delta(x) \,.</math>