Sign Function - Properties

Properties

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

From equation (1) it follows that whenever x is not equal to 0 we have

Similarly, for any real number x,

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.

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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 generalised notion of differentiation in distribution theory, the derivative of the signum function is two times the Dirac delta function, which can be demonstrated using the identity

(where H(x) is the Heaviside step function using the standard H(0) = 1/2 formalism). Using this identity, it is easy to derive the distributional derivative:

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

For, a smooth approximation of the sign function is

Another approximation is

which gets sharper as, note that it's the derivative of . This is inspired from the fact that the above is exactly equal for all nonzero x if, 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.