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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").



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


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