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### Signum function

The sign function extracts the sign of a real number *x* and is defined as

To avoid confusion with the sine function, this function is often called the signum function (from *signum*, latin for "sign").

##### Explanation

Any real number can be expressed as the product of its absolute value and its sign function

x= sgn (x)·|x|

From this equation it follows that whenever *x *≠ 0 we have

The signum function is the derivative of the absolute value function (up to the indeterminacy at zero). Note, the resultant power of *x* is 0, similar to the ordinary derivative of *x*. The numbers cancel and all we are left with is the sign of *x*.

The signum function is differentiable with derivative 0 everywhere except at 0. It is not differentiable at 0 in the ordinary sense, but under the generalized notion of differentiation in distribution theory, the derivative of the signum function is two times the Dirac delta function

##### Figure

The signum curve. The value zero is in the origin.