Univalent function: Difference between revisions

Revision as of 04:46, 9 January 2014

Revision as of 04:47, 9 January 2014

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:<math>f: (-1, 1) \to (-1, 1) \, </math>

given by ''&fnof;''(''x'')&nbsp;=&nbsp;''x''³. This function is clearly one-to-oneinjective, however,but its derivative is 0 at ''x''&nbsp;=&nbsp;0, and its inverse is not analytic, or even differentiable, on the whole interval&nbsp;(&minus;1,&nbsp;1). Consequently, if we enlarge the ___domain to an open subset ''G'' of the complex plane, it must fail to be one-to-one; and this is the case, since (for example) ''f''(&epsilon;&omega;)&nbsp;= ''f''(&epsilon;) (where &omega; is a [[primitive root of unity|primitive cube root of unity]] and &epsilon; is a positive real number smaller than the radius of ''G'' as a neighbourhood of 0).

== References==