Talk:Even and odd functions: Difference between revisions

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Revision as of 18:33, 12 May 2012

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I think for Complex function you have to use the conjugate.--[[User:Drazick|Royi A]] ([[User talk:Drazick|talk]]) 20:12, 24 September 2009 (UTC)

== Functions which are simultaneously odd and even ==

As there is no restriction for odd functions to go through the origin it's not strictly true that the only function which is both odd and even is the zero function, however it is the only continuous function to do so. Consider the characteristic function:

<math>\chi_0:[-1,1] \to \{0,1\}</math> given by <math>\chi (0) = 1</math> and <math>\chi(x)=0, x \neq 0</math>

then, for <math>x \neq 0</math> we have <math>\chi(x)=0=\chi(-x)</math> and, for <math>x=0</math> , <math>\chi(-x)=\chi(0)=0=-0=-\chi(0)</math>

However I do believe that the only real functions which are both odd and even are real multiples of the characteristic function of <math>\{0\}</math>.