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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.