Editing Sign function (section)

=== Integration ===

The signum function has a [[definite integral]] between any pair of finite values {{mvar|a}} and {{mvar|b}}, even when the interval of integration includes zero. The resulting integral for {{mvar|a}} and {{mvar|b}} is then equal to the difference between their absolute values:
<math display="block"> \int_a^b (\sgn x) \, \text{d}x = |b| - |a| \,.</math>

In fact, the signum function is the derivative of the absolute value function, except where there is an abrupt change in [[slope|gradient]] at zero:
<math display="block"> \frac{\text{d} |x|}{\text{d}x} =  \sgn x \qquad \text{for } x \ne 0\,.</math>

We can understand this as before by considering the definition of the absolute value <math>|x|</math> on the separate regions <math>x>0</math> and <math>x<0.</math> For example, the absolute value function is identical to <math>x</math> in the region <math>x>0,</math> whose derivative is the constant value {{math|+1}}, which equals the value of <math>\sgn x</math> there.

Because the absolute value is a [[convex function]], there is at least one [[subderivative]] at every point, including at the origin. Everywhere except zero, the resulting [[subdifferential]] consists of a single value, equal to the value of the sign function. In contrast, there are many subderivatives at zero, with just one of them taking the value <math>\sgn(0) = 0</math>. A subderivative value {{math|0}} occurs here because the absolute value function is at a minimum. The full family of valid subderivatives at zero constitutes the subdifferential interval <math>[-1,1]</math>, which might be thought of informally as "filling in" the graph of the sign function with a vertical line through the origin, making it continuous as a two dimensional curve.

In integration theory, the signum function is a [[weak derivative]] of the absolute value function. Weak derivatives are equivalent if they are equal [[almost everywhere]], making them impervious to isolated anomalies at a single point. This includes the change in gradient of the absolute value function at zero, which prohibits there being a classical derivative.