Sign function

In mathematical notation the sign function is often represented as

The law of trichotomy states that every real number must be positive, negative or zero.

The signum function denotes which unique category a number falls into by mapping it to one of the values −1, +1 or 0, which can then be used in mathematical expressions or further calculations.

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

The signum can also be written using the Iverson bracket notation:

The signum can also be written using the floor and the absolute value functions:

is accepted to be equal to 1, the signum can also be written for all real numbers as

Either jump demonstrates visually that the sign function

These observations are confirmed by any of the various equivalent formal definitions of continuity in mathematical analysis.

make up any infinite sequence which becomes arbitrarily close to

The arrow symbol can be read to mean approaches, or tends to, and it applies to the sequence as a whole.

Despite the sign function having a very simple form, the step change at zero causes difficulties for traditional calculus techniques, which are quite stringent in their requirements.

, and has the advantage of simple generalization to higher-dimensional analogues of the sign function (for example, the partial derivatives of

See Heaviside step function § Analytic approximations.

This follows from the differentiability of any constant function, for which the derivative is always zero on its domain of definition.

acts as a constant function when it is restricted to the negative open region

It can similarly be regarded as a constant function within the positive open region

Although these are two different constant functions, their derivative is equal to zero in each case.

in the ordinary sense, under the generalized notion of differentiation in distribution theory, the derivative of the signum function is two times the Dirac delta function.

The resulting integral for a and b is then equal to the difference between their absolute values:

Because the absolute value is a convex function, there is at least one subderivative at every point, including at the origin.

The full family of valid subderivatives at zero constitutes the subdifferential interval

, 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.

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.

means taking the Cauchy principal value.

is the point on the unit circle of the complex plane that is nearest to

For reasons of symmetry, and to keep this a proper generalization of the signum function on the reals, also in the complex domain one usually defines, for

Another generalization of the sign function for real and complex expressions is

In particular, the generalized signum anticommutes with the Dirac delta function[6]