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### Signum

The **signum 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= |x| · sgn (x)

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

For *x* ≠ 0, the signum function is the derivative of the absolute value. Except for *x* ≠ 0, the signum function is differentiable with derivative 0 everywhere.

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

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*.

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**