Editing Sign function (section)

=== Signum as a generalized function ===
<!-- Was "Generalized signum function" -->

At real values of <math>x</math>, it is possible to define a [[generalized function]]&ndash;version of the signum function, <math>\varepsilon (x)</math> such that <math>\varepsilon (x)^2=1</math> everywhere, including at the point <math>x=0</math>, unlike <math>\sgn</math>, for which <math>(\sgn 0)^2=0</math>. This generalized signum allows construction of the [[algebra of generalized functions]], but the price of such generalization is the loss of [[commutativity]]. In particular, the generalized signum anticommutes with the Dirac delta function<ref name="Algebra">
{{cite journal
 |author  = Yu.M.Shirokov
 |title   = Algebra of one-dimensional generalized functions
 |journal = [[Theoretical and Mathematical Physics]]
 |year    = 1979
 |volume  = 39
 |issue   = 3
 |pages   = 471–477
 |url     = http://springerlink.metapress.com/content/w3010821x8267824/?p=5bb23f98d846495c808e0a2e642b983a&pi=3
 |archive-url = https://archive.today/20121208232109/http://springerlink.metapress.com/content/w3010821x8267824/?p=5bb23f98d846495c808e0a2e642b983a&pi=3
 |url-status = dead
 |archive-date = 2012-12-08
 |doi     = 10.1007/BF01017992
|bibcode = 1979TMP....39..471S
 }}</ref>
<math display="block">\varepsilon (x) \delta(x)+\delta(x) \varepsilon(x) = 0 \, ;</math>
in addition, <math>\varepsilon (x)</math> cannot be evaluated at <math>x=0</math>; and the special name, <math>\varepsilon</math> is necessary to distinguish it from the function <math>\sgn</math>. (<math>\varepsilon (0)</math> is not defined, but <math>\sgn 0=0</math>.)